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## G = C23×SL2(𝔽3)  order 192 = 26·3

### Direct product of C23 and SL2(𝔽3)

Aliases: C23×SL2(𝔽3), C24.12A4, Q8⋊(C22×C6), (Q8×C23)⋊1C3, C2.3(C23×A4), (C22×Q8)⋊5C6, C23.31(C2×A4), C22.18(C22×A4), (C2×Q8)⋊9(C2×C6), SmallGroup(192,1498)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C23×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C22×SL2(𝔽3) — C23×SL2(𝔽3)
 Lower central Q8 — C23×SL2(𝔽3)
 Upper central C1 — C24

Generators and relations for C23×SL2(𝔽3)
G = < a,b,c,d,e,f | a2=b2=c2=d4=f3=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=d-1, fdf-1=e, fef-1=de >

Subgroups: 709 in 280 conjugacy classes, 99 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, Q8, Q8, C23, C2×C6, C22×C4, C2×Q8, C2×Q8, C24, SL2(𝔽3), C22×C6, C23×C4, C22×Q8, C22×Q8, C2×SL2(𝔽3), C23×C6, Q8×C23, C22×SL2(𝔽3), C23×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C6, C23, A4, C2×C6, SL2(𝔽3), C2×A4, C22×C6, C2×SL2(𝔽3), C22×A4, C22×SL2(𝔽3), C23×A4, C23×SL2(𝔽3)

Smallest permutation representation of C23×SL2(𝔽3)
On 64 points
Generators in S64
(1 42)(2 43)(3 44)(4 41)(5 35)(6 36)(7 33)(8 34)(9 39)(10 40)(11 37)(12 38)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)
(1 36)(2 33)(3 34)(4 35)(5 41)(6 42)(7 43)(8 44)(9 29)(10 30)(11 31)(12 32)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)(37 63)(38 64)(39 61)(40 62)(45 55)(46 56)(47 53)(48 54)(49 59)(50 60)(51 57)(52 58)
(1 20)(2 17)(3 18)(4 19)(5 57)(6 58)(7 59)(8 60)(9 13)(10 14)(11 15)(12 16)(21 31)(22 32)(23 29)(24 30)(25 35)(26 36)(27 33)(28 34)(37 47)(38 48)(39 45)(40 46)(41 51)(42 52)(43 49)(44 50)(53 63)(54 64)(55 61)(56 62)
(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 12 3 10)(2 11 4 9)(5 61 7 63)(6 64 8 62)(13 17 15 19)(14 20 16 18)(21 25 23 27)(22 28 24 26)(29 33 31 35)(30 36 32 34)(37 41 39 43)(38 44 40 42)(45 49 47 51)(46 52 48 50)(53 57 55 59)(54 60 56 58)
(2 11 12)(4 9 10)(5 61 62)(7 63 64)(13 14 19)(15 16 17)(21 22 27)(23 24 25)(29 30 35)(31 32 33)(37 38 43)(39 40 41)(45 46 51)(47 48 49)(53 54 59)(55 56 57)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 3A 3B 4A ··· 4H 6A ··· 6AD order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 4 4 6 ··· 6 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 2 2 3 3 type + + - + + image C1 C2 C3 C6 SL2(𝔽3) SL2(𝔽3) A4 C2×A4 kernel C23×SL2(𝔽3) C22×SL2(𝔽3) Q8×C23 C22×Q8 C23 C23 C24 C23 # reps 1 7 2 14 8 16 1 7

Matrix representation of C23×SL2(𝔽3) in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 9 3 0 0 0 3 4
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 1 0
,
 9 0 0 0 0 0 9 0 0 0 0 0 3 0 0 0 0 0 1 9 0 0 0 0 3

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,3,0,0,0,3,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,0],[9,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,9,3] >;

C23×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_2^3\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2^3xSL(2,3)");
// GroupNames label

G:=SmallGroup(192,1498);
// by ID

G=gap.SmallGroup(192,1498);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,235,172,404,285,124]);
// Polycyclic

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

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