Copied to
clipboard

## G = C22×S3≀C2order 288 = 25·32

### Direct product of C22 and S3≀C2

Series: Derived Chief Lower central Upper central

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

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 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 3A 3B 4A 4B 4C 4D 6A ··· 6F 6G ··· 6N order 1 2 2 2 2 ··· 2 2 2 2 2 3 3 4 4 4 4 6 ··· 6 6 ··· 6 size 1 1 1 1 6 ··· 6 9 9 9 9 4 4 18 18 18 18 4 ··· 4 12 ··· 12

36 irreducible representations

 dim 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 D4 D4 S3≀C2 C2×S3≀C2 kernel C22×S3≀C2 C2×S3≀C2 C22×C32⋊C4 C22×S32 C2×C3⋊S3 C62 C22 C2 # reps 1 12 1 2 3 1 4 12

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

 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 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 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 -1 -1 -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 0 -1 1 0 0 0 0 -1 -1 -2 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 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

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

׿
×
𝔽