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G = C2×C32.3S4  order 432 = 24·33

Direct product of C2 and C32.3S4

direct product, non-abelian, soluble, monomial

Aliases: C2×C32.3S4, C62.51D6, C6⋊(C3.S4), (C2×C6)⋊3D18, C3.A4⋊3D6, C6.8(C3⋊S4), (C3×C6).18S4, (C22×C6)⋊2D9, C23⋊2(C9⋊S3), C32.4(C2×S4), (C2×C62).16S3, C3⋊2(C2×C3.S4), C3.2(C2×C3⋊S4), (C2×C3.A4)⋊2S3, (C6×C3.A4)⋊3C2, C22⋊2(C2×C9⋊S3), (C3×C3.A4)⋊4C22, (C22×C6).7(C3⋊S3), (C2×C6).3(C2×C3⋊S3), SmallGroup(432,537)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×C3.A4 — C2×C32.3S4
Chief series
C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C32.3S4 — C2×C32.3S4
Lower central
C3×C3.A4 — C2×C32.3S4
Upper central
C1 — C2

Generators and relations for C2×C32.3S4
 G = < a,b,c,d,e | a2=b6=c6=e2=1, d3=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b4c3, ebe=b2c3, dcd-1=b3c, ece=b3c2, ede=c4d2 >

Subgroups: 1540 in 178 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, C32, Dic3, D6, C2×C6, C2×C6, C2×C6, C2×D4, D9, C18, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C3×C9, C3.A4, D18, C3⋊Dic3, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C9⋊S3, C3×C18, C3.S4, C2×C3.A4, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C2×C62, C3×C3.A4, C2×C9⋊S3, C2×C3.S4, C2×C32⋊7D4, C32.3S4, C6×C3.A4, C2×C32.3S4
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, S4, D18, C2×C3⋊S3, C2×S4, C9⋊S3, C3.S4, C3⋊S4, C2×C9⋊S3, C2×C3.S4, C2×C3⋊S4, C32.3S4, C2×C32.3S4

Smallest permutation representation of C2×C32.3S4
►On 54 points
Generators in S54
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)

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

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

(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)

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B6A6B6C6D6E···6L9A···9I18A···18I
order12222233334466666···69···918···18
size113354542222545422226···68···88···8

42 irreducible representations

dim111222222336666
type+++++++++++++++
imageC1C2C2S3S3D6D6D9D18S4C2×S4C3.S4C3⋊S4C2×C3.S4C2×C3⋊S4
kernelC2×C32.3S4C32.3S4C6×C3.A4C2×C3.A4C2×C62C3.A4C62C22×C6C2×C6C3×C6C32C6C6C3C3
# reps121313199223131

Matrix representation of C2×C32.3S4 ►in GL7(𝔽37)

36000000
03600000
00360000
00036000
00003600
00000360
00000036
,
363600000
1000000
0010000
0001000
00003600
00000360
00003601
,
1000000
0100000
00036000
00136000
00003600
00003610
00000036
,
363600000
1000000
002631000
00620000
00001035
00000036
00000136
,
1000000
363600000
0001000
0010000
0000100
0000001
0000010

G:=sub<GL(7,GF(37))| [36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[36,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,36,0,0,0,0,0,36,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,36,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36],[36,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,26,6,0,0,0,0,0,31,20,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,35,36,36],[1,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;
 

C2×C32.3S4 in GAP, Magma, Sage, TeX 

C_2\times C_3^2._3S_4
 
% in TeX
 
G:=Group("C2xC3^2.3S4");
 
// GroupNames label
 
G:=SmallGroup(432,537);
 
// by ID
 
G=gap.SmallGroup(432,537);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,926,394,675,2524,9077,2287,5298,3989]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^6=c^6=e^2=1,d^3=c^2,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=c^4*d^2>;
 
// generators/relations
 

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