Copied to
clipboard

## G = C2×C32⋊S4order 432 = 24·33

### Direct product of C2 and C32⋊S4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C32⋊S4
G = < a,b,c,d,e | a2=b6=c6=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b4c, ebe=b2c3, dcd-1=b3c, ece=b3c4, ede=d-1 >

Subgroups: 967 in 166 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×4], C3, C3 [×4], C4 [×2], C22, C22 [×6], S3 [×8], C6, C6 [×12], C2×C4, D4 [×4], C23, C23, C32, C32 [×3], Dic3 [×2], C12 [×2], A4 [×3], D6 [×7], C2×C6, C2×C6 [×11], C2×D4, C3×S3 [×8], C3×C6, C3×C6 [×5], C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×4], S4 [×6], C2×A4 [×3], C22×S3, C22×C6, C22×C6 [×2], He3, C3×Dic3 [×2], C3×A4 [×3], S3×C6 [×7], C62, C62 [×2], C2×C3⋊D4, C6×D4, C2×S4 [×3], He3⋊C2 [×2], C2×He3, C6×Dic3, C3×C3⋊D4 [×4], C3×S4 [×6], C6×A4 [×3], S3×C2×C6, C2×C62, C32⋊A4, C2×He3⋊C2, C6×C3⋊D4, C6×S4 [×3], C32⋊S4 [×2], C2×C32⋊A4, C2×C32⋊S4
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], C3⋊S3, S4, C2×C3⋊S3, C2×S4, He3⋊C2, C3⋊S4, C2×He3⋊C2, C2×C3⋊S4, C32⋊S4, C2×C32⋊S4

Permutation representations of C2×C32⋊S4
On 18 points - transitive group 18T156
Generators in S18
(1 2)(3 4)(5 6)(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 10 8 11 9 12)(13 17 15)(14 18 16)
(1 16 12)(2 13 8)(3 14 11)(4 17 7)(5 18 10)(6 15 9)
(7 17)(8 13)(9 15)(10 18)(11 14)(12 16)

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

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

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

G:=TransitiveGroup(18,156);

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 6G ··· 6M 6N 6O 6P 6Q 6R 6S 6T 12A 12B 12C 12D order 1 2 2 2 2 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 size 1 1 3 3 18 18 1 1 6 24 24 24 18 18 1 1 3 3 3 3 6 ··· 6 18 18 18 18 24 24 24 18 18 18 18

38 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 3 3 3 3 6 6 6 6 type + + + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 S4 C2×S4 He3⋊C2 C2×He3⋊C2 C32⋊S4 C2×C32⋊S4 C3⋊S4 C2×C3⋊S4 C32⋊S4 C2×C32⋊S4 kernel C2×C32⋊S4 C32⋊S4 C2×C32⋊A4 C6×A4 C2×C62 C3×A4 C62 C3×C6 C32 C23 C22 C2 C1 C6 C3 C2 C1 # reps 1 2 1 3 1 3 1 2 2 4 4 4 4 1 1 2 2

Matrix representation of C2×C32⋊S4 in GL3(𝔽7) generated by

 6 0 0 0 6 0 0 0 6
,
 4 3 3 1 1 0 4 1 3
,
 1 6 1 6 3 2 2 4 1
,
 5 4 1 6 0 6 4 4 2
,
 1 0 5 0 1 5 0 0 6
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,1,4,3,1,1,3,0,3],[1,6,2,6,3,4,1,2,1],[5,6,4,4,0,4,1,6,2],[1,0,0,0,1,0,5,5,6] >;

C2×C32⋊S4 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes S_4
% in TeX

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

G:=SmallGroup(432,538);
// by ID

G=gap.SmallGroup(432,538);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,170,675,353,9077,2287,5298,3989]);
// 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,e*b*e=b^2*c^3,d*c*d^-1=b^3*c,e*c*e=b^3*c^4,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽