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G = C22×S3≀C2order 288 = 25·32

Direct product of C22 and S3≀C2

direct product, non-abelian, soluble, monomial, rational

Aliases: C22×S3≀C2, C623D4, S32⋊C23, C32⋊C4⋊C23, C32⋊(C22×D4), C3⋊S3.1C24, C3⋊S3⋊(C2×D4), (C3×C6)⋊(C2×D4), (C2×C3⋊S3)⋊8D4, (C22×S32)⋊10C2, (C2×S32)⋊13C22, (C22×C32⋊C4)⋊5C2, (C2×C32⋊C4)⋊4C22, (C2×C3⋊S3).32C23, (C22×C3⋊S3).60C22, SmallGroup(288,1031)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C22×S3≀C2
C1C32C3⋊S3S32S3≀C2C2×S3≀C2 — C22×S3≀C2
C32C3⋊S3 — C22×S3≀C2
C1C22

Generators and relations for C22×S3≀C2
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=d, fcf=ede-1=c-1, df=fd, fef=e-1 >

Subgroups: 1952 in 370 conjugacy classes, 83 normal (8 characteristic)
C1, C2 [×3], C2 [×12], C3 [×2], C4 [×4], C22, C22 [×38], S3 [×16], C6 [×14], C2×C4 [×6], D4 [×16], C23 [×21], C32, D6 [×56], C2×C6 [×14], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×8], C3⋊S3, C3⋊S3 [×3], C3×C6 [×3], C22×S3 [×28], C22×C6 [×2], C22×D4, C32⋊C4 [×4], S32 [×8], S32 [×12], S3×C6 [×12], C2×C3⋊S3 [×6], C62, S3×C23 [×2], S3≀C2 [×16], C2×C32⋊C4 [×6], C2×S32 [×12], C2×S32 [×6], S3×C2×C6 [×2], C22×C3⋊S3, C2×S3≀C2 [×12], C22×C32⋊C4, C22×S32 [×2], C22×S3≀C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, S3≀C2, C2×S3≀C2 [×3], C22×S3≀C2

Permutation representations of C22×S3≀C2
On 24 points - transitive group 24T654
Generators in S24
(1 4)(2 3)(5 7)(6 8)(9 19)(10 20)(11 17)(12 18)(13 23)(14 24)(15 21)(16 22)
(1 6)(2 5)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(2 12 10)(3 18 20)(5 14 16)(7 24 22)
(1 11 9)(4 17 19)(6 13 15)(8 23 21)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 5)(4 6)(9 21)(10 24)(11 23)(12 22)(13 17)(14 20)(15 19)(16 18)

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

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

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

G:=TransitiveGroup(24,654);

On 24 points - transitive group 24T657
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 24)(10 21)(11 22)(12 23)(13 19)(14 20)(15 17)(16 18)
(1 3)(2 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 23)
(1 20 21)(2 22 17)(3 23 18)(4 19 24)(5 11 15)(6 12 16)(7 13 9)(8 14 10)
(1 21 20)(2 22 17)(3 18 23)(4 19 24)(5 11 15)(6 16 12)(7 13 9)(8 10 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 2)(3 4)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 22)(23 24)

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

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

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

G:=TransitiveGroup(24,657);

36 conjugacy classes

class 1 2A2B2C2D···2K2L2M2N2O3A3B4A4B4C4D6A···6F6G···6N
order12222···222223344446···66···6
size11116···6999944181818184···412···12

36 irreducible representations

dim11112244
type++++++++
imageC1C2C2C2D4D4S3≀C2C2×S3≀C2
kernelC22×S3≀C2C2×S3≀C2C22×C32⋊C4C22×S32C2×C3⋊S3C62C22C2
# reps1121231412

Matrix representation of C22×S3≀C2 in GL8(ℤ)

10000000
01000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00000-100
00001-100
00000110
00000101
,
10000000
01000000
00100000
00010000
00001000
00000100
0000-1-1-1-1
00000010
,
0-1000000
10000000
00010000
00-100000
000000-11
0000-1-1-2-1
00000010
00000110
,
-10000000
01000000
00100000
000-10000
00000100
00001000
00000010
00000001

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,-1,-2,1,1,0,0,0,0,1,-1,0,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C22×S3≀C2 in GAP, Magma, Sage, TeX

C_2^2\times S_3\wr C_2
% in TeX

G:=Group("C2^2xS3wrC2");
// GroupNames label

G:=SmallGroup(288,1031);
// by ID

G=gap.SmallGroup(288,1031);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,253,2693,2028,201,797,622]);
// Polycyclic

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

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