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## G = C2×C32⋊4S4order 432 = 24·33

### Direct product of C2 and C32⋊4S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×A4 — C2×C32⋊4S4
 Chief series C1 — C22 — C2×C6 — C62 — C32×A4 — C32⋊4S4 — C2×C32⋊4S4
 Lower central C32×A4 — C2×C32⋊4S4
 Upper central C1 — C2

Generators and relations for C2×C324S4
G = < a,b,c,d,e | a2=b6=c6=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b4c3, ebe=b2c3, dcd-1=b3c, ece=b3c2, ede=d-1 >

Subgroups: 3340 in 358 conjugacy classes, 71 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, C32, Dic3, A4, D6, C2×C6, C2×C6, C2×D4, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, S4, C2×A4, C22×S3, C22×C6, C33, C3⋊Dic3, C3×A4, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C2×S4, C33⋊C2, C32×C6, C2×C3⋊Dic3, C327D4, C3⋊S4, C6×A4, C22×C3⋊S3, C2×C62, C32×A4, C2×C33⋊C2, C2×C327D4, C2×C3⋊S4, C324S4, A4×C3×C6, C2×C324S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, C2×S4, C33⋊C2, C3⋊S4, C2×C33⋊C2, C2×C3⋊S4, C324S4, C2×C324S4

Smallest permutation representation of C2×C324S4
On 54 points
Generators in S54
(1 7)(2 8)(3 9)(4 11)(5 12)(6 10)(13 17)(14 18)(15 16)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 52)(50 53)(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 15 4 7 16 11)(2 13 5 8 17 12)(3 14 6 9 18 10)(19 26 44)(20 27 45)(21 28 46)(22 29 47)(23 30 48)(24 25 43)(31 53 39 34 50 42)(32 54 40 35 51 37)(33 49 41 36 52 38)
(1 40 21)(2 38 19)(3 42 23)(4 51 46)(5 49 44)(6 53 48)(7 37 24)(8 41 22)(9 39 20)(10 50 45)(11 54 43)(12 52 47)(13 33 29)(14 31 27)(15 35 25)(16 32 28)(17 36 26)(18 34 30)
(1 47)(2 43)(3 45)(4 22)(5 24)(6 20)(7 44)(8 46)(9 48)(10 23)(11 19)(12 21)(13 28)(14 30)(15 26)(16 29)(17 25)(18 27)(31 34)(32 33)(35 36)(37 49)(38 54)(39 53)(40 52)(41 51)(42 50)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E ··· 3M 4A 4B 6A 6B 6C 6D 6E ··· 6L 6M ··· 6U order 1 2 2 2 2 2 3 3 3 3 3 ··· 3 4 4 6 6 6 6 6 ··· 6 6 ··· 6 size 1 1 3 3 54 54 2 2 2 2 8 ··· 8 54 54 2 2 2 2 6 ··· 6 8 ··· 8

42 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 S4 C2×S4 C3⋊S4 C2×C3⋊S4 kernel C2×C32⋊4S4 C32⋊4S4 A4×C3×C6 C6×A4 C2×C62 C3×A4 C62 C3×C6 C32 C6 C3 # reps 1 2 1 12 1 12 1 2 2 4 4

Matrix representation of C2×C324S4 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
,
 0 -1 0 0 0 0 0 1 -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
,
 0 -1 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 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 -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 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 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 0 -1 0 0 0 0 0 -1 0

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],[0,1,0,0,0,0,0,-1,-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],[0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-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],[0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[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,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0] >;

C2×C324S4 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_4S_4
% in TeX

G:=Group("C2xC3^2:4S4");
// GroupNames label

G:=SmallGroup(432,762);
// by ID

G=gap.SmallGroup(432,762);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,170,675,2524,9077,2287,5298,3989]);
// Polycyclic

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

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