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## G = C8×SL2(𝔽3)  order 192 = 26·3

### Direct product of C8 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C8×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — C2×Q8 — C4×Q8 — C4×SL2(𝔽3) — C8×SL2(𝔽3)
 Lower central Q8 — C8×SL2(𝔽3)
 Upper central C1 — C2×C8

Generators and relations for C8×SL2(𝔽3)
G = < a,b,c,d | a8=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Smallest permutation representation of C8×SL2(𝔽3)
On 64 points
Generators in S64
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 57 40 48)(2 58 33 41)(3 59 34 42)(4 60 35 43)(5 61 36 44)(6 62 37 45)(7 63 38 46)(8 64 39 47)(9 54 25 24)(10 55 26 17)(11 56 27 18)(12 49 28 19)(13 50 29 20)(14 51 30 21)(15 52 31 22)(16 53 32 23)
(1 49 40 19)(2 50 33 20)(3 51 34 21)(4 52 35 22)(5 53 36 23)(6 54 37 24)(7 55 38 17)(8 56 39 18)(9 45 25 62)(10 46 26 63)(11 47 27 64)(12 48 28 57)(13 41 29 58)(14 42 30 59)(15 43 31 60)(16 44 32 61)
(9 54 62)(10 55 63)(11 56 64)(12 49 57)(13 50 58)(14 51 59)(15 52 60)(16 53 61)(17 46 26)(18 47 27)(19 48 28)(20 41 29)(21 42 30)(22 43 31)(23 44 32)(24 45 25)

G:=sub<Sym(64)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,57,40,48)(2,58,33,41)(3,59,34,42)(4,60,35,43)(5,61,36,44)(6,62,37,45)(7,63,38,46)(8,64,39,47)(9,54,25,24)(10,55,26,17)(11,56,27,18)(12,49,28,19)(13,50,29,20)(14,51,30,21)(15,52,31,22)(16,53,32,23), (1,49,40,19)(2,50,33,20)(3,51,34,21)(4,52,35,22)(5,53,36,23)(6,54,37,24)(7,55,38,17)(8,56,39,18)(9,45,25,62)(10,46,26,63)(11,47,27,64)(12,48,28,57)(13,41,29,58)(14,42,30,59)(15,43,31,60)(16,44,32,61), (9,54,62)(10,55,63)(11,56,64)(12,49,57)(13,50,58)(14,51,59)(15,52,60)(16,53,61)(17,46,26)(18,47,27)(19,48,28)(20,41,29)(21,42,30)(22,43,31)(23,44,32)(24,45,25)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,57,40,48)(2,58,33,41)(3,59,34,42)(4,60,35,43)(5,61,36,44)(6,62,37,45)(7,63,38,46)(8,64,39,47)(9,54,25,24)(10,55,26,17)(11,56,27,18)(12,49,28,19)(13,50,29,20)(14,51,30,21)(15,52,31,22)(16,53,32,23), (1,49,40,19)(2,50,33,20)(3,51,34,21)(4,52,35,22)(5,53,36,23)(6,54,37,24)(7,55,38,17)(8,56,39,18)(9,45,25,62)(10,46,26,63)(11,47,27,64)(12,48,28,57)(13,41,29,58)(14,42,30,59)(15,43,31,60)(16,44,32,61), (9,54,62)(10,55,63)(11,56,64)(12,49,57)(13,50,58)(14,51,59)(15,52,60)(16,53,61)(17,46,26)(18,47,27)(19,48,28)(20,41,29)(21,42,30)(22,43,31)(23,44,32)(24,45,25) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,57,40,48),(2,58,33,41),(3,59,34,42),(4,60,35,43),(5,61,36,44),(6,62,37,45),(7,63,38,46),(8,64,39,47),(9,54,25,24),(10,55,26,17),(11,56,27,18),(12,49,28,19),(13,50,29,20),(14,51,30,21),(15,52,31,22),(16,53,32,23)], [(1,49,40,19),(2,50,33,20),(3,51,34,21),(4,52,35,22),(5,53,36,23),(6,54,37,24),(7,55,38,17),(8,56,39,18),(9,45,25,62),(10,46,26,63),(11,47,27,64),(12,48,28,57),(13,41,29,58),(14,42,30,59),(15,43,31,60),(16,44,32,61)], [(9,54,62),(10,55,63),(11,56,64),(12,49,57),(13,50,58),(14,51,59),(15,52,60),(16,53,61),(17,46,26),(18,47,27),(19,48,28),(20,41,29),(21,42,30),(22,43,31),(23,44,32),(24,45,25)]])

56 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 8A ··· 8H 8I 8J 8K 8L 12A ··· 12H 24A ··· 24P order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 4 4 1 1 1 1 6 6 6 6 4 ··· 4 1 ··· 1 6 6 6 6 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + - + + image C1 C2 C3 C4 C6 C8 C12 C24 SL2(𝔽3) SL2(𝔽3) C4.A4 C8.A4 A4 C2×A4 C4×A4 C8×A4 kernel C8×SL2(𝔽3) C4×SL2(𝔽3) C8×Q8 C2×SL2(𝔽3) C4×Q8 SL2(𝔽3) C2×Q8 Q8 C8 C8 C4 C2 C2×C8 C2×C4 C22 C2 # reps 1 1 2 2 2 4 4 8 2 4 6 12 1 1 2 4

Matrix representation of C8×SL2(𝔽3) in GL3(𝔽73) generated by

 51 0 0 0 51 0 0 0 51
,
 1 0 0 0 64 8 0 8 9
,
 1 0 0 0 0 72 0 1 0
,
 64 0 0 0 1 64 0 0 8
G:=sub<GL(3,GF(73))| [51,0,0,0,51,0,0,0,51],[1,0,0,0,64,8,0,8,9],[1,0,0,0,0,1,0,72,0],[64,0,0,0,1,0,0,64,8] >;

C8×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_8\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C8xSL(2,3)");
// GroupNames label

G:=SmallGroup(192,200);
// by ID

G=gap.SmallGroup(192,200);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,2,-2,42,58,851,172,1524,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations

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