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## G = C2×D4×C3⋊S3order 288 = 25·32

### Direct product of C2, D4 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×D4×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C23×C3⋊S3 — C2×D4×C3⋊S3
 Lower central C32 — C3×C6 — C2×D4×C3⋊S3
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4×C3⋊S3
G = < a,b,c,d,e,f | a2=b4=c2=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 2916 in 708 conjugacy classes, 173 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×12], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], S3 [×32], C6 [×12], C6 [×16], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], C32, Dic3 [×8], C12 [×8], D6 [×120], C2×C6 [×20], C2×C6 [×16], C22×C4, C2×D4, C2×D4 [×11], C24 [×2], C3⋊S3 [×4], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×4], C4×S3 [×16], D12 [×16], C2×Dic3 [×4], C3⋊D4 [×32], C2×C12 [×4], C3×D4 [×16], C22×S3 [×76], C22×C6 [×8], C22×D4, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×10], C2×C3⋊S3 [×20], C62, C62 [×4], C62 [×4], S3×C2×C4 [×4], C2×D12 [×4], S3×D4 [×32], C2×C3⋊D4 [×8], C6×D4 [×4], S3×C23 [×8], C4×C3⋊S3 [×4], C12⋊S3 [×4], C2×C3⋊Dic3, C327D4 [×8], C6×C12, D4×C32 [×4], C22×C3⋊S3, C22×C3⋊S3 [×10], C22×C3⋊S3 [×8], C2×C62 [×2], C2×S3×D4 [×4], C2×C4×C3⋊S3, C2×C12⋊S3, D4×C3⋊S3 [×8], C2×C327D4 [×2], D4×C3×C6, C23×C3⋊S3 [×2], C2×D4×C3⋊S3
Quotients: C1, C2 [×15], C22 [×35], S3 [×4], D4 [×4], C23 [×15], D6 [×28], C2×D4 [×6], C24, C3⋊S3, C22×S3 [×28], C22×D4, C2×C3⋊S3 [×7], S3×D4 [×8], S3×C23 [×4], C22×C3⋊S3 [×7], C2×S3×D4 [×4], D4×C3⋊S3 [×2], C23×C3⋊S3, C2×D4×C3⋊S3

Smallest permutation representation of C2×D4×C3⋊S3
On 72 points
Generators in S72
(1 41)(2 42)(3 43)(4 44)(5 23)(6 24)(7 21)(8 22)(9 36)(10 33)(11 34)(12 35)(13 37)(14 38)(15 39)(16 40)(17 46)(18 47)(19 48)(20 45)(25 69)(26 70)(27 71)(28 72)(29 68)(30 65)(31 66)(32 67)(49 62)(50 63)(51 64)(52 61)(53 59)(54 60)(55 57)(56 58)
(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 44)(2 43)(3 42)(4 41)(5 24)(6 23)(7 22)(8 21)(9 35)(10 34)(11 33)(12 36)(13 40)(14 39)(15 38)(16 37)(17 45)(18 48)(19 47)(20 46)(25 72)(26 71)(27 70)(28 69)(29 65)(30 68)(31 67)(32 66)(49 63)(50 62)(51 61)(52 64)(53 60)(54 59)(55 58)(56 57)
(1 37 19)(2 38 20)(3 39 17)(4 40 18)(5 28 35)(6 25 36)(7 26 33)(8 27 34)(9 24 69)(10 21 70)(11 22 71)(12 23 72)(13 48 41)(14 45 42)(15 46 43)(16 47 44)(29 64 55)(30 61 56)(31 62 53)(32 63 54)(49 59 66)(50 60 67)(51 57 68)(52 58 65)
(1 24 32)(2 21 29)(3 22 30)(4 23 31)(5 66 44)(6 67 41)(7 68 42)(8 65 43)(9 54 19)(10 55 20)(11 56 17)(12 53 18)(13 25 50)(14 26 51)(15 27 52)(16 28 49)(33 57 45)(34 58 46)(35 59 47)(36 60 48)(37 69 63)(38 70 64)(39 71 61)(40 72 62)
(1 41)(2 42)(3 43)(4 44)(5 31)(6 32)(7 29)(8 30)(9 50)(10 51)(11 52)(12 49)(13 19)(14 20)(15 17)(16 18)(21 68)(22 65)(23 66)(24 67)(25 54)(26 55)(27 56)(28 53)(33 64)(34 61)(35 62)(36 63)(37 48)(38 45)(39 46)(40 47)(57 70)(58 71)(59 72)(60 69)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 6M ··· 6AB 12A ··· 12H order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 9 9 9 9 18 18 18 18 2 2 2 2 2 2 18 18 2 ··· 2 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 S3×D4 kernel C2×D4×C3⋊S3 C2×C4×C3⋊S3 C2×C12⋊S3 D4×C3⋊S3 C2×C32⋊7D4 D4×C3×C6 C23×C3⋊S3 C6×D4 C2×C3⋊S3 C2×C12 C3×D4 C22×C6 C6 # reps 1 1 1 8 2 1 2 4 4 4 16 8 8

Matrix representation of C2×D4×C3⋊S3 in GL6(ℤ)

 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×D4×C3⋊S3 in GAP, Magma, Sage, TeX

C_2\times D_4\times C_3\rtimes S_3
% in TeX

G:=Group("C2xD4xC3:S3");
// GroupNames label

G:=SmallGroup(288,1007);
// by ID

G=gap.SmallGroup(288,1007);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,185,2693,9414]);
// Polycyclic

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

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