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## G = C2×A4×C3⋊S3order 432 = 24·33

### Direct product of C2, A4 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C2×A4×C3⋊S3
 Chief series C1 — C3 — C32 — C62 — C32×A4 — A4×C3⋊S3 — C2×A4×C3⋊S3
 Lower central C62 — C2×A4×C3⋊S3
 Upper central C1 — C2

Generators and relations for C2×A4×C3⋊S3
G = < a,b,c,d,e,f,g | a2=b2=c2=d3=e3=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, dbd-1=bc=cb, be=eb, bf=fb, bg=gb, dcd-1=b, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, geg=e-1, gfg=f-1 >

Subgroups: 1768 in 266 conjugacy classes, 45 normal (15 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C23, C32, C32, A4, A4, D6, C2×C6, C2×C6, C24, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C2×A4, C2×A4, C22×S3, C22×C6, C33, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C22×A4, S3×C23, C3×C3⋊S3, C32×C6, S3×A4, C6×A4, C6×A4, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C32×A4, C6×C3⋊S3, C2×S3×A4, C23×C3⋊S3, A4×C3⋊S3, A4×C3×C6, C2×A4×C3⋊S3
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, C3⋊S3, C2×A4, S3×C6, C2×C3⋊S3, C22×A4, C3×C3⋊S3, S3×A4, C6×C3⋊S3, C2×S3×A4, A4×C3⋊S3, C2×A4×C3⋊S3

Smallest permutation representation of C2×A4×C3⋊S3
On 54 points
Generators in S54
(1 8)(2 9)(3 7)(4 42)(5 40)(6 41)(10 13)(11 14)(12 15)(16 20)(17 21)(18 19)(22 25)(23 26)(24 27)(28 31)(29 32)(30 33)(34 38)(35 39)(36 37)(43 47)(44 48)(45 46)(49 52)(50 53)(51 54)
(1 8)(2 9)(4 42)(6 41)(11 14)(12 15)(17 21)(18 19)(23 26)(24 27)(29 32)(30 33)(35 39)(36 37)(43 47)(44 48)(50 53)(51 54)
(2 9)(3 7)(4 42)(5 40)(10 13)(12 15)(16 20)(18 19)(22 25)(24 27)(28 31)(30 33)(34 38)(36 37)(44 48)(45 46)(49 52)(51 54)
(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)
(1 50 11)(2 51 12)(3 49 10)(4 37 44)(5 38 45)(6 39 43)(7 52 13)(8 53 14)(9 54 15)(16 28 22)(17 29 23)(18 30 24)(19 33 27)(20 31 25)(21 32 26)(34 46 40)(35 47 41)(36 48 42)
(1 23 41)(2 24 42)(3 22 40)(4 9 27)(5 7 25)(6 8 26)(10 28 46)(11 29 47)(12 30 48)(13 31 45)(14 32 43)(15 33 44)(16 34 49)(17 35 50)(18 36 51)(19 37 54)(20 38 52)(21 39 53)
(4 27)(5 25)(6 26)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 46)(17 47)(18 48)(19 44)(20 45)(21 43)(22 40)(23 41)(24 42)(28 34)(29 35)(30 36)(31 38)(32 39)(33 37)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G ··· 3N 6A 6B 6C 6D 6E 6F 6G ··· 6N 6O ··· 6V 6W 6X 6Y 6Z order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 ··· 3 6 6 6 6 6 6 6 ··· 6 6 ··· 6 6 6 6 6 size 1 1 3 3 9 9 27 27 2 2 2 2 4 4 8 ··· 8 2 2 2 2 4 4 6 ··· 6 8 ··· 8 36 36 36 36

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 A4 C2×A4 C2×A4 S3×A4 C2×S3×A4 kernel C2×A4×C3⋊S3 A4×C3⋊S3 A4×C3×C6 C23×C3⋊S3 C22×C3⋊S3 C2×C62 C6×A4 C3×A4 C22×C6 C2×C6 C2×C3⋊S3 C3⋊S3 C3×C6 C6 C3 # reps 1 2 1 2 4 2 4 4 8 8 1 2 1 4 4

Matrix representation of C2×A4×C3⋊S3 in GL7(ℤ)

 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 -1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 -2 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

G:=sub<GL(7,Integers())| [-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,-2,1],[0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

C2×A4×C3⋊S3 in GAP, Magma, Sage, TeX

C_2\times A_4\times C_3\rtimes S_3
% in TeX

G:=Group("C2xA4xC3:S3");
// GroupNames label

G:=SmallGroup(432,764);
// by ID

G=gap.SmallGroup(432,764);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-3,269,123,4037,14118]);
// Polycyclic

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

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