Aliases: C33⋊12M4(2), C62.13Dic3, C33⋊4C8⋊9C2, (C3×C62).4C4, C3⋊Dic3.43D6, C3⋊Dic3.8Dic3, C22.(C33⋊C4), C3⋊2(C62.C4), C32⋊7(C4.Dic3), C6.13(C2×C32⋊C4), (C2×C6).5(C32⋊C4), (C3×C3⋊Dic3).7C4, C2.6(C2×C33⋊C4), (C2×C3⋊Dic3).11S3, (C6×C3⋊Dic3).11C2, (C32×C6).20(C2×C4), (C3×C6).27(C2×Dic3), (C3×C3⋊Dic3).51C22, SmallGroup(432,640)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊12M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=ab-1, ae=ea, bc=cb, dbd-1=a-1b-1, be=eb, dcd-1=c-1, ce=ec, ede=d5 >
Subgroups: 392 in 92 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C62, C62, C4.Dic3, C32×C6, C32×C6, C32⋊2C8, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C3×C62, C62.C4, C33⋊4C8, C6×C3⋊Dic3, C33⋊12M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), C2×Dic3, C32⋊C4, C4.Dic3, C2×C32⋊C4, C33⋊C4, C62.C4, C2×C33⋊C4, C33⋊12M4(2)
►On 24 points - transitive group 24T1287
(1 22 10)(2 11 23)(3 12 24)(4 17 13)(5 18 14)(6 15 19)(7 16 20)(8 21 9)
(1 10 22)(3 24 12)(5 14 18)(7 20 16)
(1 22 10)(2 11 23)(3 24 12)(4 13 17)(5 18 14)(6 15 19)(7 20 16)(8 9 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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)
G:=sub<Sym(24)| (1,22,10)(2,11,23)(3,12,24)(4,17,13)(5,18,14)(6,15,19)(7,16,20)(8,21,9), (1,10,22)(3,24,12)(5,14,18)(7,20,16), (1,22,10)(2,11,23)(3,24,12)(4,13,17)(5,18,14)(6,15,19)(7,20,16)(8,9,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)>;
G:=Group( (1,22,10)(2,11,23)(3,12,24)(4,17,13)(5,18,14)(6,15,19)(7,16,20)(8,21,9), (1,10,22)(3,24,12)(5,14,18)(7,20,16), (1,22,10)(2,11,23)(3,24,12)(4,13,17)(5,18,14)(6,15,19)(7,20,16)(8,9,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23) );
G=PermutationGroup([[(1,22,10),(2,11,23),(3,12,24),(4,17,13),(5,18,14),(6,15,19),(7,16,20),(8,21,9)], [(1,10,22),(3,24,12),(5,14,18),(7,20,16)], [(1,22,10),(2,11,23),(3,24,12),(4,13,17),(5,18,14),(6,15,19),(7,20,16),(8,9,21)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23)]])
G:=TransitiveGroup(24,1287);
42 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | ··· | 3G | 4A | 4B | 4C | 6A | 6B | 6C | 6D | ··· | 6U | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 9 | 9 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 54 | 54 | 54 | 54 | 18 | 18 | 18 | 18 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | C4.Dic3 | C32⋊C4 | C2×C32⋊C4 | C33⋊C4 | C62.C4 | C2×C33⋊C4 | C33⋊12M4(2) |
| kernel | C33⋊12M4(2) | C33⋊4C8 | C6×C3⋊Dic3 | C3×C3⋊Dic3 | C3×C62 | C2×C3⋊Dic3 | C3⋊Dic3 | C3⋊Dic3 | C62 | C33 | C32 | C2×C6 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C33⋊12M4(2) ►in GL4(𝔽7) generated by
| 3 | 2 | 4 | 3 |
| 4 | 5 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 1 |
| 0 | 5 | 2 | 6 |
| 0 | 2 | 0 | 2 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 4 |
| 3 | 6 | 3 | 2 |
| 6 | 3 | 4 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 6 | 1 | 0 |
| 3 | 3 | 1 | 4 |
| 4 | 3 | 5 | 5 |
| 2 | 2 | 5 | 3 |
| 0 | 6 | 3 | 2 |
| 6 | 0 | 4 | 2 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[3,3,4,2,6,3,3,2,1,1,5,5,0,4,5,3],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1] >;
C33⋊12M4(2) in GAP, Magma, Sage, TeX
C_3^3\rtimes_{12}M_4(2) % in TeX
G:=Group("C3^3:12M4(2)"); // GroupNames label
G:=SmallGroup(432,640);
// by ID
G=gap.SmallGroup(432,640);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,58,2804,298,2693,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,a*e=e*a,b*c=c*b,d*b*d^-1=a^-1*b^-1,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^5>;
// generators/relations