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## G = C2×C6.S4order 288 = 25·32

### Direct product of C2 and C6.S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3.A4 — C2×C6.S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4 — C6.S4 — C2×C6.S4
 Lower central C3.A4 — C2×C6.S4
 Upper central C1 — C22

Generators and relations for C2×C6.S4
G = < a,b,c,d,e,f | a2=b6=c2=d2=1, e3=b2, f2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=b4e2 >

Subgroups: 498 in 110 conjugacy classes, 28 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, C9, Dic3, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C18, C2×Dic3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, Dic9, C3.A4, C2×C18, C6.D4, C22×Dic3, C23×C6, C2×Dic9, C2×C3.A4, C2×C3.A4, C2×C6.D4, C6.S4, C22×C3.A4, C2×C6.S4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, D9, C2×Dic3, S4, Dic9, D18, A4⋊C4, C2×S4, C2×Dic9, C3.S4, C2×A4⋊C4, C6.S4, C2×C3.S4, C2×C6.S4

Smallest permutation representation of C2×C6.S4
On 72 points
Generators in S72
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 64)(9 65)(10 54)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 37)(28 59)(29 60)(30 61)(31 62)(32 63)(33 55)(34 56)(35 57)(36 58)
(1 62 4 56 7 59)(2 63 5 57 8 60)(3 55 6 58 9 61)(10 25 13 19 16 22)(11 26 14 20 17 23)(12 27 15 21 18 24)(28 66 31 69 34 72)(29 67 32 70 35 64)(30 68 33 71 36 65)(37 50 40 53 43 47)(38 51 41 54 44 48)(39 52 42 46 45 49)
(1 38)(2 46)(3 58)(4 41)(5 49)(6 61)(7 44)(8 52)(9 55)(10 34)(11 67)(12 21)(13 28)(14 70)(15 24)(16 31)(17 64)(18 27)(19 66)(20 35)(22 69)(23 29)(25 72)(26 32)(30 71)(33 65)(36 68)(37 53)(39 57)(40 47)(42 60)(43 50)(45 63)(48 59)(51 62)(54 56)
(1 56)(2 39)(3 47)(4 59)(5 42)(6 50)(7 62)(8 45)(9 53)(10 19)(11 35)(12 68)(13 22)(14 29)(15 71)(16 25)(17 32)(18 65)(20 67)(21 36)(23 70)(24 30)(26 64)(27 33)(28 69)(31 72)(34 66)(37 55)(38 54)(40 58)(41 48)(43 61)(44 51)(46 57)(49 60)(52 63)
(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 65 66 67 68 69 70 71 72)
(1 54 56 38)(2 53 57 37)(3 52 58 45)(4 51 59 44)(5 50 60 43)(6 49 61 42)(7 48 62 41)(8 47 63 40)(9 46 55 39)(10 34 19 66)(11 33 20 65)(12 32 21 64)(13 31 22 72)(14 30 23 71)(15 29 24 70)(16 28 25 69)(17 36 26 68)(18 35 27 67)

G:=sub<Sym(72)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,64)(9,65)(10,54)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,37)(28,59)(29,60)(30,61)(31,62)(32,63)(33,55)(34,56)(35,57)(36,58), (1,62,4,56,7,59)(2,63,5,57,8,60)(3,55,6,58,9,61)(10,25,13,19,16,22)(11,26,14,20,17,23)(12,27,15,21,18,24)(28,66,31,69,34,72)(29,67,32,70,35,64)(30,68,33,71,36,65)(37,50,40,53,43,47)(38,51,41,54,44,48)(39,52,42,46,45,49), (1,38)(2,46)(3,58)(4,41)(5,49)(6,61)(7,44)(8,52)(9,55)(10,34)(11,67)(12,21)(13,28)(14,70)(15,24)(16,31)(17,64)(18,27)(19,66)(20,35)(22,69)(23,29)(25,72)(26,32)(30,71)(33,65)(36,68)(37,53)(39,57)(40,47)(42,60)(43,50)(45,63)(48,59)(51,62)(54,56), (1,56)(2,39)(3,47)(4,59)(5,42)(6,50)(7,62)(8,45)(9,53)(10,19)(11,35)(12,68)(13,22)(14,29)(15,71)(16,25)(17,32)(18,65)(20,67)(21,36)(23,70)(24,30)(26,64)(27,33)(28,69)(31,72)(34,66)(37,55)(38,54)(40,58)(41,48)(43,61)(44,51)(46,57)(49,60)(52,63), (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)(64,65,66,67,68,69,70,71,72), (1,54,56,38)(2,53,57,37)(3,52,58,45)(4,51,59,44)(5,50,60,43)(6,49,61,42)(7,48,62,41)(8,47,63,40)(9,46,55,39)(10,34,19,66)(11,33,20,65)(12,32,21,64)(13,31,22,72)(14,30,23,71)(15,29,24,70)(16,28,25,69)(17,36,26,68)(18,35,27,67)>;

G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,64)(9,65)(10,54)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,37)(28,59)(29,60)(30,61)(31,62)(32,63)(33,55)(34,56)(35,57)(36,58), (1,62,4,56,7,59)(2,63,5,57,8,60)(3,55,6,58,9,61)(10,25,13,19,16,22)(11,26,14,20,17,23)(12,27,15,21,18,24)(28,66,31,69,34,72)(29,67,32,70,35,64)(30,68,33,71,36,65)(37,50,40,53,43,47)(38,51,41,54,44,48)(39,52,42,46,45,49), (1,38)(2,46)(3,58)(4,41)(5,49)(6,61)(7,44)(8,52)(9,55)(10,34)(11,67)(12,21)(13,28)(14,70)(15,24)(16,31)(17,64)(18,27)(19,66)(20,35)(22,69)(23,29)(25,72)(26,32)(30,71)(33,65)(36,68)(37,53)(39,57)(40,47)(42,60)(43,50)(45,63)(48,59)(51,62)(54,56), (1,56)(2,39)(3,47)(4,59)(5,42)(6,50)(7,62)(8,45)(9,53)(10,19)(11,35)(12,68)(13,22)(14,29)(15,71)(16,25)(17,32)(18,65)(20,67)(21,36)(23,70)(24,30)(26,64)(27,33)(28,69)(31,72)(34,66)(37,55)(38,54)(40,58)(41,48)(43,61)(44,51)(46,57)(49,60)(52,63), (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)(64,65,66,67,68,69,70,71,72), (1,54,56,38)(2,53,57,37)(3,52,58,45)(4,51,59,44)(5,50,60,43)(6,49,61,42)(7,48,62,41)(8,47,63,40)(9,46,55,39)(10,34,19,66)(11,33,20,65)(12,32,21,64)(13,31,22,72)(14,30,23,71)(15,29,24,70)(16,28,25,69)(17,36,26,68)(18,35,27,67) );

G=PermutationGroup([[(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,64),(9,65),(10,54),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,37),(28,59),(29,60),(30,61),(31,62),(32,63),(33,55),(34,56),(35,57),(36,58)], [(1,62,4,56,7,59),(2,63,5,57,8,60),(3,55,6,58,9,61),(10,25,13,19,16,22),(11,26,14,20,17,23),(12,27,15,21,18,24),(28,66,31,69,34,72),(29,67,32,70,35,64),(30,68,33,71,36,65),(37,50,40,53,43,47),(38,51,41,54,44,48),(39,52,42,46,45,49)], [(1,38),(2,46),(3,58),(4,41),(5,49),(6,61),(7,44),(8,52),(9,55),(10,34),(11,67),(12,21),(13,28),(14,70),(15,24),(16,31),(17,64),(18,27),(19,66),(20,35),(22,69),(23,29),(25,72),(26,32),(30,71),(33,65),(36,68),(37,53),(39,57),(40,47),(42,60),(43,50),(45,63),(48,59),(51,62),(54,56)], [(1,56),(2,39),(3,47),(4,59),(5,42),(6,50),(7,62),(8,45),(9,53),(10,19),(11,35),(12,68),(13,22),(14,29),(15,71),(16,25),(17,32),(18,65),(20,67),(21,36),(23,70),(24,30),(26,64),(27,33),(28,69),(31,72),(34,66),(37,55),(38,54),(40,58),(41,48),(43,61),(44,51),(46,57),(49,60),(52,63)], [(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),(64,65,66,67,68,69,70,71,72)], [(1,54,56,38),(2,53,57,37),(3,52,58,45),(4,51,59,44),(5,50,60,43),(6,49,61,42),(7,48,62,41),(8,47,63,40),(9,46,55,39),(10,34,19,66),(11,33,20,65),(12,32,21,64),(13,31,22,72),(14,30,23,71),(15,29,24,70),(16,28,25,69),(17,36,26,68),(18,35,27,67)]])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4H 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 18A ··· 18I order 1 2 2 2 2 2 2 2 3 4 ··· 4 6 6 6 6 6 6 6 9 9 9 18 ··· 18 size 1 1 1 1 3 3 3 3 2 18 ··· 18 2 2 2 6 6 6 6 8 8 8 8 ··· 8

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 6 type + + + + - + + - + + + + - + image C1 C2 C2 C4 S3 Dic3 D6 D9 Dic9 D18 S4 A4⋊C4 C2×S4 C3.S4 C6.S4 C2×C3.S4 kernel C2×C6.S4 C6.S4 C22×C3.A4 C2×C3.A4 C23×C6 C22×C6 C22×C6 C24 C23 C23 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 4 1 2 1 3 6 3 2 4 2 1 2 1

Matrix representation of C2×C6.S4 in GL5(𝔽37)

 36 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 36 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 31 6 1
,
 20 6 0 0 0 31 26 0 0 0 0 0 0 0 1 0 0 6 31 36 0 0 16 15 6
,
 29 36 0 0 0 28 8 0 0 0 0 0 0 1 0 0 0 36 0 0 0 0 22 21 6

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,1,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,31,0,0,0,36,6,0,0,0,0,1],[20,31,0,0,0,6,26,0,0,0,0,0,0,6,16,0,0,0,31,15,0,0,1,36,6],[29,28,0,0,0,36,8,0,0,0,0,0,0,36,22,0,0,1,0,21,0,0,0,0,6] >;

C2×C6.S4 in GAP, Magma, Sage, TeX

C_2\times C_6.S_4
% in TeX

G:=Group("C2xC6.S4");
// GroupNames label

G:=SmallGroup(288,341);
// by ID

G=gap.SmallGroup(288,341);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,1123,192,1684,6053,782,3534,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^6=c^2=d^2=1,e^3=b^2,f^2=b^3,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,f*b*f^-1=b^-1,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=b^4*e^2>;
// generators/relations

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