Order = 9828 = 2^{2}.3^{3}.7.13.
Mult = 2.
Out = 6.
Porting notes
Porting incomplete.Standard generators
Standard generators of L_{2}(27) are a, b where a has order 2, b has order 3 and ab has order 7.
Standard generators of 2.L_{2}(27) = SL_{2}(27) are preimages A, B where B has order 3 and AB has order 7.
Standard generators of L_{2}(27).2 are c, d where c has order 2 (necessarily in class 2B), d has order 4 and cd has order 7.
Standard generators of 2L_{2}(27).2 are preimages C, D where CD has order 7.
Standard generators of L_{2}(27).3 are e, f where e has order 2, f has order 3, ef has order 9, efeff has order 7 and efefefeffefeffeff has order 9.
Standard generators of 2L_{2}(27).3 are preimages E, F where F has order 3 and EF has order 9.
Standard generators of L_{2}(27).6 are g, h where g has order 2 (necessarily in class 2B), h has order 3 (necessarily in class 3C), gh has order 24 and ghghh has order 13.
Standard generators of 2L_{2}(27).6 are preimages G, H where H has order 3.
Black box algorithms
Checking generators (semipresentations)
Group  Semipresentation  File 

L_{2}(27)  〈〈 a, b  some conditions 〉〉  Download 
Representations
Representations of L_{2}(27)
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 Permutation representations:
Number of points ID Generators Description Link 28 Std Details  Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[ω] 13 a Std Details 0 Z[ω] 13 b Std Details 0 Z 26 g Std Details 0 Z 27 Std Details 0 C 28 a Std Details 0 C 28 b Std Details 0 C 28 c Std Details 0 C 28 d Std Details 0 C 28 e Std Details 0 C 28 f Std Details 0 Z 78 a Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 13 a Std Details 2 GF(4) 13 b Std Details Char Ring Dimension ID Generators Description Link 3 GF(27) 3 a Std Details 3 GF(3) 27 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 26 Std Details Char Ring Dimension ID Generators Description Link 13 GF(13) 27 Std Details
Representations of 2.L_{2}(27)
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 Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(27) 2 a Std Details 3 GF(27) 2 b Std Details 3 GF(27) 2 c Std Details
Conjugacy classes
Conjugacy classes of L_{2}(27)
Conjugacy class  Centraliser order  Power up  Class rep(s) 

1A  9 828  (b)^{3} 

2A  28 
(abababbabb)^{7} 

3A  27  b 

3B  27  bb 

7A  14 
abababbabbabababbabb 

7B  14 
Omitted
owing to
length. 

7C  14 
abababbabbabababbabbabababbabbabababbabb 

13A  13 
ababbababbababbababbababbababbababbababb 

13B  13 
ababbababb 

13C  13 
Omitted
owing to
length. 

13D  13 
ababb 

13E  13 
Omitted
owing to
length. 

13F  13 
ababbababbababbababb 

14A  14 
(abababbabb)^{9} 

14B  14 
abababbabb 

14C  14 
(abababbabb)^{3} 
Download words for class representatives.