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G = C2×S3×C32⋊C4order 432 = 24·33

Direct product of C2, S3 and C32⋊C4

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×S3×C32⋊C4, C331(C22×C4), C33⋊C42C22, (S3×C3×C6)⋊2C4, (S3×C3⋊S3)⋊2C4, C3⋊S35(C4×S3), (C3×C6)⋊5(C4×S3), (C32×C6)⋊(C2×C4), C61(C2×C32⋊C4), (C6×C32⋊C4)⋊6C2, C3210(S3×C2×C4), (C2×C3⋊S3).38D6, C31(C22×C32⋊C4), (S3×C32)⋊1(C2×C4), (C2×C33⋊C2)⋊2C4, C33⋊C21(C2×C4), (C2×C33⋊C4)⋊5C2, C3⋊S3.6(C22×S3), (C3×C3⋊S3).5C23, (S3×C3⋊S3).3C22, (C3×C32⋊C4)⋊3C22, (C6×C3⋊S3).32C22, (C2×S3×C3⋊S3).3C2, (C3×C3⋊S3)⋊4(C2×C4), SmallGroup(432,753)

Series: Derived Chief Lower central Upper central

C1C33 — C2×S3×C32⋊C4
C1C3C33C3×C3⋊S3S3×C3⋊S3S3×C32⋊C4 — C2×S3×C32⋊C4
C33 — C2×S3×C32⋊C4
C1C2

Generators and relations for C2×S3×C32⋊C4
 G = < a,b,c,d,e,f | a2=b3=c2=d3=e3=f4=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, fef-1=de=ed, fdf-1=d-1e >

Subgroups: 1632 in 192 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C4 [×4], C22 [×7], S3 [×2], S3 [×14], C6, C6 [×10], C2×C4 [×6], C23, C32, C32 [×4], Dic3 [×2], C12 [×2], D6, D6 [×19], C2×C6 [×3], C22×C4, C3×S3 [×8], C3⋊S3 [×2], C3⋊S3 [×10], C3×C6, C3×C6 [×6], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3 [×3], C33, C32⋊C4 [×2], C32⋊C4 [×2], S32 [×8], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×9], C62, S3×C2×C4, S3×C32 [×2], C3×C3⋊S3 [×2], C33⋊C2 [×2], C32×C6, C2×C32⋊C4, C2×C32⋊C4 [×5], C2×S32 [×2], C22×C3⋊S3, C3×C32⋊C4 [×2], C33⋊C4 [×2], S3×C3⋊S3 [×4], S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C22×C32⋊C4, S3×C32⋊C4 [×4], C6×C32⋊C4, C2×C33⋊C4, C2×S3×C3⋊S3, C2×S3×C32⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4×S3 [×2], C22×S3, C32⋊C4, S3×C2×C4, C2×C32⋊C4 [×3], C22×C32⋊C4, S3×C32⋊C4, C2×S3×C32⋊C4

Permutation representations of C2×S3×C32⋊C4
On 24 points - transitive group 24T1310
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 7)(2 8)(3 5)(4 6)(9 22)(10 23)(11 24)(12 21)(13 17)(14 18)(15 19)(16 20)
(1 21 20)(2 17 22)(3 18 23)(4 24 19)(5 14 10)(6 11 15)(7 12 16)(8 13 9)
(2 22 17)(4 19 24)(6 15 11)(8 9 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

G:=sub<Sym(24)| (1,5)(2,6)(3,7)(4,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,7)(2,8)(3,5)(4,6)(9,22)(10,23)(11,24)(12,21)(13,17)(14,18)(15,19)(16,20), (1,21,20)(2,17,22)(3,18,23)(4,24,19)(5,14,10)(6,11,15)(7,12,16)(8,13,9), (2,22,17)(4,19,24)(6,15,11)(8,9,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;

G:=Group( (1,5)(2,6)(3,7)(4,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,7)(2,8)(3,5)(4,6)(9,22)(10,23)(11,24)(12,21)(13,17)(14,18)(15,19)(16,20), (1,21,20)(2,17,22)(3,18,23)(4,24,19)(5,14,10)(6,11,15)(7,12,16)(8,13,9), (2,22,17)(4,19,24)(6,15,11)(8,9,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );

G=PermutationGroup([(1,5),(2,6),(3,7),(4,8),(9,19),(10,20),(11,17),(12,18),(13,24),(14,21),(15,22),(16,23)], [(1,20,21),(2,17,22),(3,18,23),(4,19,24),(5,10,14),(6,11,15),(7,12,16),(8,9,13)], [(1,7),(2,8),(3,5),(4,6),(9,22),(10,23),(11,24),(12,21),(13,17),(14,18),(15,19),(16,20)], [(1,21,20),(2,17,22),(3,18,23),(4,24,19),(5,14,10),(6,11,15),(7,12,16),(8,13,9)], [(2,22,17),(4,19,24),(6,15,11),(8,9,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)])

G:=TransitiveGroup(24,1310);

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G6H6I6J6K12A12B12C12D
order1222222233333444444446666666666612121212
size1133992727244889999272727272448812121212181818181818

36 irreducible representations

dim111111112222244488
type+++++++++++++
imageC1C2C2C2C2C4C4C4S3D6D6C4×S3C4×S3C32⋊C4C2×C32⋊C4C2×C32⋊C4S3×C32⋊C4C2×S3×C32⋊C4
kernelC2×S3×C32⋊C4S3×C32⋊C4C6×C32⋊C4C2×C33⋊C4C2×S3×C3⋊S3S3×C3⋊S3S3×C3×C6C2×C33⋊C2C2×C32⋊C4C32⋊C4C2×C3⋊S3C3⋊S3C3×C6D6S3C6C2C1
# reps141114221212224222

Matrix representation of C2×S3×C32⋊C4 in GL6(𝔽13)

1200000
0120000
0012000
0001200
0000120
0000012
,
1210000
1200000
001000
000100
000010
000001
,
010000
100000
001000
000100
000010
000001
,
100000
010000
00011212
001211212
00121012
00121120
,
100000
010000
00121120
001211212
00011212
00121012
,
500000
050000
008000
008050
000855
008500

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,12,0,0,1,1,1,1,0,0,12,12,0,12,0,0,12,12,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,12,0,0,1,1,1,1,0,0,12,12,12,0,0,0,0,12,12,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,8,0,0,0,0,8,5,0,0,0,5,5,0,0,0,0,0,5,0] >;

C2×S3×C32⋊C4 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_3^2\rtimes C_4
% in TeX

G:=Group("C2xS3xC3^2:C4");
// GroupNames label

G:=SmallGroup(432,753);
// by ID

G=gap.SmallGroup(432,753);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,64,1411,165,1356,530,14118]);
// Polycyclic

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

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