G = C3⋊S3⋊M4(2) order 288 = 25·32
2nd semidirect product of C3⋊S3 and M4(2) acting via M4(2)/C2×C4=C2
Aliases: (C6×C12).11C4, C3⋊S3⋊3M4(2), C62.15(C2×C4), C32⋊4(C2×M4(2)), C62.C4⋊6C2, C32⋊2C8⋊8C22, C32⋊M4(2)⋊8C2, C3⋊Dic3.31C23, C3⋊S3⋊3C8⋊8C2, (C4×C3⋊S3).17C4, C4.13(C2×C32⋊C4), (C3×C12).20(C2×C4), (C2×C4).8(C32⋊C4), C2.5(C22×C32⋊C4), C22.6(C2×C32⋊C4), (C4×C3⋊S3).97C22, (C22×C3⋊S3).16C4, C3⋊Dic3.52(C2×C4), (C3×C6).26(C22×C4), (C2×C3⋊Dic3).174C22, (C2×C4×C3⋊S3).27C2, (C2×C3⋊S3).46(C2×C4), SmallGroup(288,931)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊S3⋊M4(2)
G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, cac=a-1, dad-1=ab-1, ae=ea, cbc=b-1, dbd-1=a-1b-1, be=eb, cd=dc, ce=ec, ede=d5 >
Subgroups: 576 in 122 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, M4(2), C22×C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C32⋊2C8, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C3⋊S3⋊3C8, C32⋊M4(2), C62.C4, C2×C4×C3⋊S3, C3⋊S3⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C2×M4(2), C32⋊C4, C2×C32⋊C4, C22×C32⋊C4, C3⋊S3⋊M4(2)
►On 24 points - transitive group 24T624
(1 22 11)(3 13 24)(5 18 15)(7 9 20)
(1 22 11)(2 23 12)(3 13 24)(4 14 17)(5 18 15)(6 19 16)(7 9 20)(8 10 21)
(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)
G:=sub<Sym(24)| (1,22,11)(3,13,24)(5,18,15)(7,9,20), (1,22,11)(2,23,12)(3,13,24)(4,14,17)(5,18,15)(6,19,16)(7,9,20)(8,10,21), (9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)>;
G:=Group( (1,22,11)(3,13,24)(5,18,15)(7,9,20), (1,22,11)(2,23,12)(3,13,24)(4,14,17)(5,18,15)(6,19,16)(7,9,20)(8,10,21), (9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23) );
G=PermutationGroup([[(1,22,11),(3,13,24),(5,18,15),(7,9,20)], [(1,22,11),(2,23,12),(3,13,24),(4,14,17),(5,18,15),(6,19,16),(7,9,20),(8,10,21)], [(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23)]])
G:=TransitiveGroup(24,624);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 4 | 4 | 1 | 1 | 2 | 9 | 9 | 18 | 4 | ··· | 4 | 18 | ··· | 18 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | C32⋊C4 | C2×C32⋊C4 | C2×C32⋊C4 | C3⋊S3⋊M4(2) |
| kernel | C3⋊S3⋊M4(2) | C3⋊S3⋊3C8 | C32⋊M4(2) | C62.C4 | C2×C4×C3⋊S3 | C4×C3⋊S3 | C6×C12 | C22×C3⋊S3 | C3⋊S3 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 2 | 4 | 2 | 8 |
Matrix representation of C3⋊S3⋊M4(2) ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 3 | 0 | 0 | 4 |
| 0 | 0 | 0 | 3 |
| 0 | 4 | 1 | 0 |
| 0 | 4 | 0 | 0 |
| 3 | 0 | 0 | 4 |
| 4 | 0 | 0 | 2 |
| 0 | 4 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 |
| 4 | 0 | 0 | 2 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [0,0,0,3,0,1,0,0,0,0,1,0,3,0,0,4],[0,0,0,3,0,4,4,0,0,1,0,0,3,0,0,4],[4,0,0,0,0,4,0,0,0,1,1,0,2,0,0,1],[0,4,0,0,2,0,0,0,0,0,0,1,0,2,2,0],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1] >;
C3⋊S3⋊M4(2) in GAP, Magma, Sage, TeX
C_3\rtimes S_3\rtimes M_4(2)
% in TeX
G:=Group("C3:S3:M4(2)"); // GroupNames label
G:=SmallGroup(288,931);
// by ID
G=gap.SmallGroup(288,931);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,422,80,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=e^2=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations