direct product, non-abelian, soluble, monomial
Aliases: C4×S3≀C2, C32⋊(C4×D4), (C3×C12)⋊3D4, C3⋊Dic3⋊4D4, C6.D6.11C22, S32⋊C4⋊6C2, (C4×S32)⋊9C2, S32⋊1(C2×C4), C2.1(C2×S3≀C2), (C4×C32⋊C4)⋊7C2, C32⋊C4⋊1(C2×C4), C3⋊S3.Q8⋊4C2, (C2×S3≀C2).3C2, (C2×S32).6C22, (C3×C6).10(C2×D4), C3⋊S3.3(C4○D4), (C2×C3⋊S3).4C23, C3⋊S3.1(C22×C4), (C4×C3⋊S3).61C22, (C2×C32⋊C4).14C22, SmallGroup(288,877)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×S3≀C2
G = < a,b,c,d,e | a4=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >
Subgroups: 736 in 148 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C4×D4, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, C32⋊C4, S32, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, S3≀C2, C2×C32⋊C4, C2×S32, S32⋊C4, C3⋊S3.Q8, C4×C32⋊C4, C4×S32, C2×S3≀C2, C4×S3≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C4×D4, S3≀C2, C2×S3≀C2, C4×S3≀C2
►On 24 points - transitive group 24T643
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)
(1 24)(2 21)(3 22)(4 23)(5 20 10 15)(6 17 11 16)(7 18 12 13)(8 19 9 14)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20), (1,24)(2,21)(3,22)(4,23)(5,20,10,15)(6,17,11,16)(7,18,12,13)(8,19,9,14), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20), (1,24)(2,21)(3,22)(4,23)(5,20,10,15)(6,17,11,16)(7,18,12,13)(8,19,9,14), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20)], [(1,24),(2,21),(3,22),(4,23),(5,20,10,15),(6,17,11,16),(7,18,12,13),(8,19,9,14)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)]])
G:=TransitiveGroup(24,643);
►On 24 points - transitive group 24T650
(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 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 22 3 24)(2 23 4 21)(5 15 12 18)(6 16 9 19)(7 13 10 20)(8 14 11 17)
(1 24)(2 21)(3 22)(4 23)(5 15)(6 16)(7 13)(8 14)(9 19)(10 20)(11 17)(12 18)
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22,3,24)(2,23,4,21)(5,15,12,18)(6,16,9,19)(7,13,10,20)(8,14,11,17), (1,24)(2,21)(3,22)(4,23)(5,15)(6,16)(7,13)(8,14)(9,19)(10,20)(11,17)(12,18)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22,3,24)(2,23,4,21)(5,15,12,18)(6,16,9,19)(7,13,10,20)(8,14,11,17), (1,24)(2,21)(3,22)(4,23)(5,15)(6,16)(7,13)(8,14)(9,19)(10,20)(11,17)(12,18) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,22,3,24),(2,23,4,21),(5,15,12,18),(6,16,9,19),(7,13,10,20),(8,14,11,17)], [(1,24),(2,21),(3,22),(4,23),(5,15),(6,16),(7,13),(8,14),(9,19),(10,20),(11,17),(12,18)]])
G:=TransitiveGroup(24,650);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 4 | 4 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 18 | 18 | 18 | 18 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | S3≀C2 | C2×S3≀C2 | C4×S3≀C2 |
| kernel | C4×S3≀C2 | S32⋊C4 | C3⋊S3.Q8 | C4×C32⋊C4 | C4×S32 | C2×S3≀C2 | S3≀C2 | C3⋊Dic3 | C3×C12 | C3⋊S3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C4×S3≀C2 ►in GL4(𝔽5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 2 | 4 | 0 |
| 4 | 0 | 0 | 4 |
| 4 | 0 | 0 | 4 |
| 0 | 0 | 2 | 0 |
| 0 | 2 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 4 | 0 | 0 | 4 |
| 0 | 0 | 0 | 3 |
| 0 | 4 | 3 | 0 |
| 0 | 1 | 2 | 0 |
| 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,0,0,4,0,0,2,0,0,2,4,0,1,0,0,4],[4,0,0,1,0,0,2,0,0,2,4,0,4,0,0,0],[0,4,0,0,0,0,0,4,2,0,0,3,0,4,3,0],[0,1,0,0,1,0,0,0,2,0,0,3,0,1,2,0] >;
C4×S3≀C2 in GAP, Magma, Sage, TeX
C_4\times S_3\wr C_2
% in TeX
G:=Group("C4xS3wrC2"); // GroupNames label
G:=SmallGroup(288,877);
// by ID
G=gap.SmallGroup(288,877);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,100,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=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=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations