Copied to
clipboard

## G = C2×S32⋊C4order 288 = 25·32

### Direct product of C2 and S32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C2×S32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — S32⋊C4 — C2×S32⋊C4
 Lower central C32 — C3⋊S3 — C2×S32⋊C4
 Upper central C1 — C22

Generators and relations for C2×S32⋊C4
G = < a,b,c,d,e,f | a2=b3=c2=d3=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=fdf-1=b-1, bd=db, be=eb, fbf-1=ede=d-1, cd=dc, ce=ec, fcf-1=e, fef-1=c >

Subgroups: 1200 in 226 conjugacy classes, 43 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C22×C4, C24, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C2×C22⋊C4, C3×Dic3, C32⋊C4, S32, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, S3×C23, C6.D6, C6.D6, C6×Dic3, C2×C32⋊C4, C2×C32⋊C4, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, S32⋊C4, C2×C6.D6, C22×C32⋊C4, C22×S32, C2×S32⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, S3≀C2, S32⋊C4, C2×S3≀C2, C2×S32⋊C4

Permutation representations of C2×S32⋊C4
On 24 points - transitive group 24T645
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)
(2 22 17)(4 24 19)(6 15 11)(8 13 9)
(1 5)(2 6)(3 7)(4 8)(9 24)(10 20)(11 22)(12 18)(13 19)(14 21)(15 17)(16 23)
(1 20 21)(3 18 23)(5 10 14)(7 12 16)
(1 5)(2 6)(3 7)(4 8)(9 19)(10 21)(11 17)(12 23)(13 24)(14 20)(15 22)(16 18)
(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), (2,22,17)(4,24,19)(6,15,11)(8,13,9), (1,5)(2,6)(3,7)(4,8)(9,24)(10,20)(11,22)(12,18)(13,19)(14,21)(15,17)(16,23), (1,20,21)(3,18,23)(5,10,14)(7,12,16), (1,5)(2,6)(3,7)(4,8)(9,19)(10,21)(11,17)(12,23)(13,24)(14,20)(15,22)(16,18), (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), (2,22,17)(4,24,19)(6,15,11)(8,13,9), (1,5)(2,6)(3,7)(4,8)(9,24)(10,20)(11,22)(12,18)(13,19)(14,21)(15,17)(16,23), (1,20,21)(3,18,23)(5,10,14)(7,12,16), (1,5)(2,6)(3,7)(4,8)(9,19)(10,21)(11,17)(12,23)(13,24)(14,20)(15,22)(16,18), (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)], [(2,22,17),(4,24,19),(6,15,11),(8,13,9)], [(1,5),(2,6),(3,7),(4,8),(9,24),(10,20),(11,22),(12,18),(13,19),(14,21),(15,17),(16,23)], [(1,20,21),(3,18,23),(5,10,14),(7,12,16)], [(1,5),(2,6),(3,7),(4,8),(9,19),(10,21),(11,17),(12,23),(13,24),(14,20),(15,22),(16,18)], [(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(24,645);

On 24 points - transitive group 24T651
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)
(2 22 17)(4 24 19)(6 15 11)(8 13 9)
(1 5)(2 8)(3 7)(4 6)(9 22)(10 20)(11 24)(12 18)(13 17)(14 21)(15 19)(16 23)
(1 20 21)(3 18 23)(5 10 14)(7 12 16)
(1 7)(2 6)(3 5)(4 8)(9 19)(10 23)(11 17)(12 21)(13 24)(14 18)(15 22)(16 20)
(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), (2,22,17)(4,24,19)(6,15,11)(8,13,9), (1,5)(2,8)(3,7)(4,6)(9,22)(10,20)(11,24)(12,18)(13,17)(14,21)(15,19)(16,23), (1,20,21)(3,18,23)(5,10,14)(7,12,16), (1,7)(2,6)(3,5)(4,8)(9,19)(10,23)(11,17)(12,21)(13,24)(14,18)(15,22)(16,20), (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), (2,22,17)(4,24,19)(6,15,11)(8,13,9), (1,5)(2,8)(3,7)(4,6)(9,22)(10,20)(11,24)(12,18)(13,17)(14,21)(15,19)(16,23), (1,20,21)(3,18,23)(5,10,14)(7,12,16), (1,7)(2,6)(3,5)(4,8)(9,19)(10,23)(11,17)(12,21)(13,24)(14,18)(15,22)(16,20), (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)], [(2,22,17),(4,24,19),(6,15,11),(8,13,9)], [(1,5),(2,8),(3,7),(4,6),(9,22),(10,20),(11,24),(12,18),(13,17),(14,21),(15,19),(16,23)], [(1,20,21),(3,18,23),(5,10,14),(7,12,16)], [(1,7),(2,6),(3,5),(4,8),(9,19),(10,23),(11,17),(12,21),(13,24),(14,18),(15,22),(16,20)], [(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(24,651);

On 24 points - transitive group 24T661
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)
(2 17 22)(4 19 24)(5 11 15)(7 9 13)
(2 5)(4 7)(9 24)(11 22)(13 19)(15 17)
(1 21 20)(3 23 18)(6 16 12)(8 14 10)
(1 8)(3 6)(10 21)(12 23)(14 20)(16 18)
(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)| (1,8)(2,5)(3,6)(4,7)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (2,17,22)(4,19,24)(5,11,15)(7,9,13), (2,5)(4,7)(9,24)(11,22)(13,19)(15,17), (1,21,20)(3,23,18)(6,16,12)(8,14,10), (1,8)(3,6)(10,21)(12,23)(14,20)(16,18), (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( (1,8)(2,5)(3,6)(4,7)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (2,17,22)(4,19,24)(5,11,15)(7,9,13), (2,5)(4,7)(9,24)(11,22)(13,19)(15,17), (1,21,20)(3,23,18)(6,16,12)(8,14,10), (1,8)(3,6)(10,21)(12,23)(14,20)(16,18), (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([[(1,8),(2,5),(3,6),(4,7),(9,19),(10,20),(11,17),(12,18),(13,24),(14,21),(15,22),(16,23)], [(2,17,22),(4,19,24),(5,11,15),(7,9,13)], [(2,5),(4,7),(9,24),(11,22),(13,19),(15,17)], [(1,21,20),(3,23,18),(6,16,12),(8,14,10)], [(1,8),(3,6),(10,21),(12,23),(14,20),(16,18)], [(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(24,661);

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D4 S3≀C2 S32⋊C4 S32⋊C4 C2×S3≀C2 kernel C2×S32⋊C4 S32⋊C4 C2×C6.D6 C22×C32⋊C4 C22×S32 C2×S32 C2×C3⋊S3 C62 C22 C2 C2 C2 # reps 1 4 1 1 1 8 3 1 4 4 4 4

Matrix representation of C2×S32⋊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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 10 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 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 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 3 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 5 12 0 0 0 0 11 8 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 1 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,10,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,3,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,11,0,0,0,0,12,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0] >;

C2×S32⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xS3^2:C4");
// GroupNames label

G:=SmallGroup(288,880);
// by ID

G=gap.SmallGroup(288,880);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,2693,2028,362,797,1203]);
// Polycyclic

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

׿
×
𝔽