metabelian, supersoluble, monomial
Aliases: C3⋊C8.22D6, C3⋊3(C8○D12), C12.42(C4×S3), C32⋊6(C8○D4), C12⋊S3.2C4, C3⋊3(D12.C4), (C2×C12).290D6, C4.Dic3⋊11S3, C62.47(C2×C4), C32⋊7D4.2C4, C32⋊4Q8.2C4, (C6×C12).68C22, C4.4(C6.D6), C12.31D6⋊11C2, C12.29D6⋊10C2, (C3×C12).146C23, C12.145(C22×S3), C12.59D6.4C2, C22.1(C6.D6), (C2×C3⋊C8)⋊4S3, (C6×C3⋊C8)⋊18C2, C4.92(C2×S32), (C2×C4).63S32, C6.26(S3×C2×C4), (C2×C6).18(C4×S3), (C3×C12).58(C2×C4), (C3×C3⋊C8).27C22, C2.4(C2×C6.D6), (C4×C3⋊S3).59C22, (C3×C4.Dic3)⋊11C2, C3⋊Dic3.19(C2×C4), (C3×C6).42(C22×C4), (C2×C3⋊S3).15(C2×C4), SmallGroup(288,465)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊C8.22D6
G = < a,b,c,d | a3=b8=1, c6=b2, d2=b4, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=b5, dcd-1=b4c5 >
Subgroups: 506 in 144 conjugacy classes, 52 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C8○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C4○D12, C3×C3⋊C8, C3×C3⋊C8, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C8○D12, D12.C4, C12.29D6, C12.31D6, C6×C3⋊C8, C3×C4.Dic3, C12.59D6, C3⋊C8.22D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, C8○D4, S32, S3×C2×C4, C6.D6, C2×S32, C8○D12, D12.C4, C2×C6.D6, C3⋊C8.22D6
►On 48 points
(1 9 17)(2 18 10)(3 11 19)(4 20 12)(5 13 21)(6 22 14)(7 15 23)(8 24 16)(25 41 33)(26 34 42)(27 43 35)(28 36 44)(29 45 37)(30 38 46)(31 47 39)(32 40 48)
(1 43 7 25 13 31 19 37)(2 32 8 38 14 44 20 26)(3 45 9 27 15 33 21 39)(4 34 10 40 16 46 22 28)(5 47 11 29 17 35 23 41)(6 36 12 42 18 48 24 30)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 15 13 3)(2 8 14 20)(4 18 16 6)(5 11 17 23)(7 21 19 9)(10 24 22 12)(25 27 37 39)(26 44 38 32)(28 30 40 42)(29 47 41 35)(31 33 43 45)(34 36 46 48)
G:=sub<Sym(48)| (1,9,17)(2,18,10)(3,11,19)(4,20,12)(5,13,21)(6,22,14)(7,15,23)(8,24,16)(25,41,33)(26,34,42)(27,43,35)(28,36,44)(29,45,37)(30,38,46)(31,47,39)(32,40,48), (1,43,7,25,13,31,19,37)(2,32,8,38,14,44,20,26)(3,45,9,27,15,33,21,39)(4,34,10,40,16,46,22,28)(5,47,11,29,17,35,23,41)(6,36,12,42,18,48,24,30), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,15,13,3)(2,8,14,20)(4,18,16,6)(5,11,17,23)(7,21,19,9)(10,24,22,12)(25,27,37,39)(26,44,38,32)(28,30,40,42)(29,47,41,35)(31,33,43,45)(34,36,46,48)>;
G:=Group( (1,9,17)(2,18,10)(3,11,19)(4,20,12)(5,13,21)(6,22,14)(7,15,23)(8,24,16)(25,41,33)(26,34,42)(27,43,35)(28,36,44)(29,45,37)(30,38,46)(31,47,39)(32,40,48), (1,43,7,25,13,31,19,37)(2,32,8,38,14,44,20,26)(3,45,9,27,15,33,21,39)(4,34,10,40,16,46,22,28)(5,47,11,29,17,35,23,41)(6,36,12,42,18,48,24,30), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,15,13,3)(2,8,14,20)(4,18,16,6)(5,11,17,23)(7,21,19,9)(10,24,22,12)(25,27,37,39)(26,44,38,32)(28,30,40,42)(29,47,41,35)(31,33,43,45)(34,36,46,48) );
G=PermutationGroup([[(1,9,17),(2,18,10),(3,11,19),(4,20,12),(5,13,21),(6,22,14),(7,15,23),(8,24,16),(25,41,33),(26,34,42),(27,43,35),(28,36,44),(29,45,37),(30,38,46),(31,47,39),(32,40,48)], [(1,43,7,25,13,31,19,37),(2,32,8,38,14,44,20,26),(3,45,9,27,15,33,21,39),(4,34,10,40,16,46,22,28),(5,47,11,29,17,35,23,41),(6,36,12,42,18,48,24,30)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,15,13,3),(2,8,14,20),(4,18,16,6),(5,11,17,23),(7,21,19,9),(10,24,22,12),(25,27,37,39),(26,44,38,32),(28,30,40,42),(29,47,41,35),(31,33,43,45),(34,36,46,48)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | ··· | 12F | 12G | ··· | 12K | 24A | ··· | 24H | 24I | 24J | 24K | 24L |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 4 | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | S32 | C6.D6 | C2×S32 | C6.D6 | D12.C4 | C3⋊C8.22D6 |
| kernel | C3⋊C8.22D6 | C12.29D6 | C12.31D6 | C6×C3⋊C8 | C3×C4.Dic3 | C12.59D6 | C32⋊4Q8 | C12⋊S3 | C32⋊7D4 | C2×C3⋊C8 | C4.Dic3 | C3⋊C8 | C2×C12 | C12 | C2×C6 | C32 | C3 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 4 |
Matrix representation of C3⋊C8.22D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 27 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 46 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,22,0,0,0,0,51,0,0,0,0,0,0,0,46,0,0,0,0,0,27,27],[0,27,0,0,0,0,46,27,0,0,0,0,0,0,22,0,0,0,0,0,0,51,0,0,0,0,0,0,27,0,0,0,0,0,46,46],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;
C3⋊C8.22D6 in GAP, Magma, Sage, TeX
C_3\rtimes C_8._{22}D_6 % in TeX
G:=Group("C3:C8.22D6"); // GroupNames label
G:=SmallGroup(288,465);
// by ID
G=gap.SmallGroup(288,465);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,219,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=1,c^6=b^2,d^2=b^4,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^4*c^5>;
// generators/relations