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G = C2×S32⋊C4order 288 = 25·32

Direct product of C2 and S32⋊C4

direct product, non-abelian, soluble, monomial

Aliases: C2×S32⋊C4, C62.8D4, C22.12S3≀C2, C6.D613C22, (C2×S32)⋊4C4, S322(C2×C4), C32⋊(C2×C22⋊C4), C2.2(C2×S3≀C2), C3⋊S3⋊(C22⋊C4), (C3×C6)⋊(C22⋊C4), C3⋊S3.5(C2×D4), (C2×C3⋊S3).34D4, (C22×S32).3C2, (C2×S32).8C22, (C3×C6).13(C2×D4), (C2×C3⋊S3).7C23, C3⋊S3.4(C22×C4), (C22×C32⋊C4)⋊1C2, (C2×C32⋊C4)⋊2C22, (C2×C6.D6)⋊16C2, (C22×C3⋊S3).49C22, (C2×C3⋊S3).17(C2×C4), SmallGroup(288,880)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C2×S32⋊C4
C1C32C3⋊S3C2×C3⋊S3C2×S32S32⋊C4 — C2×S32⋊C4
C32C3⋊S3 — C2×S32⋊C4
C1C22

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 [×2], C2 [×8], C3 [×2], C4 [×4], C22, C22 [×22], S3 [×12], C6 [×10], C2×C4 [×8], C23 [×11], C32, Dic3 [×2], C12 [×2], D6 [×34], C2×C6 [×8], C22⋊C4 [×4], C22×C4 [×2], C24, C3×S3 [×4], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3 [×15], C22×C6, C2×C22⋊C4, C3×Dic3 [×2], C32⋊C4 [×2], S32 [×4], S32 [×6], S3×C6 [×6], C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62, S3×C2×C4, S3×C23, C6.D6 [×2], C6.D6, C6×Dic3, C2×C32⋊C4 [×2], C2×C32⋊C4 [×2], C2×S32 [×6], C2×S32 [×3], S3×C2×C6, C22×C3⋊S3, S32⋊C4 [×4], C2×C6.D6, C22×C32⋊C4, C22×S32, C2×S32⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, S3≀C2, S32⋊C4 [×2], 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 2A2B2C2D2E2F2G2H2I2J2K3A3B4A4B4C4D4E4F4G4H6A···6F6G6H6I6J12A12B12C12D
order12222222222233444444446···6666612121212
size111166669999446666181818184···41212121212121212

36 irreducible representations

dim111111224444
type++++++++++
imageC1C2C2C2C2C4D4D4S3≀C2S32⋊C4S32⋊C4C2×S3≀C2
kernelC2×S32⋊C4S32⋊C4C2×C6.D6C22×C32⋊C4C22×S32C2×S32C2×C3⋊S3C62C22C2C2C2
# reps141118314444

Matrix representation of C2×S32⋊C4 in GL6(𝔽13)

1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
0012100
0012000
000010
000001
,
100000
10120000
000100
001000
0000120
0000012
,
100000
010000
001000
000100
0000012
0000112
,
1200000
310000
0012000
0001200
000001
000010
,
5120000
1180000
0000120
0000012
001000
000100

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

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