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## G = C2×S3×C32⋊C4order 432 = 24·33

### Direct product of C2, S3 and C32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×S3×C32⋊C4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — S3×C3⋊S3 — S3×C32⋊C4 — C2×S3×C32⋊C4
 Lower central C33 — C2×S3×C32⋊C4
 Upper central C1 — C2

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 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 size 1 1 3 3 9 9 27 27 2 4 4 8 8 9 9 9 9 27 27 27 27 2 4 4 8 8 12 12 12 12 18 18 18 18 18 18

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 C4×S3 C4×S3 C32⋊C4 C2×C32⋊C4 C2×C32⋊C4 S3×C32⋊C4 C2×S3×C32⋊C4 kernel C2×S3×C32⋊C4 S3×C32⋊C4 C6×C32⋊C4 C2×C33⋊C4 C2×S3×C3⋊S3 S3×C3⋊S3 S3×C3×C6 C2×C33⋊C2 C2×C32⋊C4 C32⋊C4 C2×C3⋊S3 C3⋊S3 C3×C6 D6 S3 C6 C2 C1 # reps 1 4 1 1 1 4 2 2 1 2 1 2 2 2 4 2 2 2

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

 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 1 0 0 0 0 12 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 1 12 12 0 0 12 1 12 12 0 0 12 1 0 12 0 0 12 1 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 12 0 0 0 12 1 12 12 0 0 0 1 12 12 0 0 12 1 0 12
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 8 0 5 0 0 0 0 8 5 5 0 0 8 5 0 0

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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