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 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×6], C6 [×2], C6 [×4], C8 [×4], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×6], C12 [×4], C12 [×2], D6 [×6], C2×C6 [×2], C2×C6, C2×C8 [×3], M4(2) [×3], C4○D4, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8 [×2], C24 [×4], Dic6 [×3], C4×S3 [×6], D12 [×3], C3⋊D4 [×6], C2×C12 [×2], C2×C12, C8○D4, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C62, S3×C8 [×4], C8⋊S3 [×4], C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C4○D12 [×3], C3×C3⋊C8 [×2], C3×C3⋊C8 [×2], C32⋊4Q8, C4×C3⋊S3 [×2], C12⋊S3, C32⋊7D4 [×2], C6×C12, C8○D12, D12.C4, C12.29D6 [×2], C12.31D6 [×2], C6×C3⋊C8, C3×C4.Dic3, C12.59D6, C3⋊C8.22D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], C8○D4, S32, S3×C2×C4 [×2], C6.D6 [×2], 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 33 41)(26 42 34)(27 35 43)(28 44 36)(29 37 45)(30 46 38)(31 39 47)(32 48 40)
(1 36 7 42 13 48 19 30)(2 25 8 31 14 37 20 43)(3 38 9 44 15 26 21 32)(4 27 10 33 16 39 22 45)(5 40 11 46 17 28 23 34)(6 29 12 35 18 41 24 47)
(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 43 37 31)(26 36 38 48)(27 29 39 41)(28 46 40 34)(30 32 42 44)(33 35 45 47)
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)(25,33,41)(26,42,34)(27,35,43)(28,44,36)(29,37,45)(30,46,38)(31,39,47)(32,48,40), (1,36,7,42,13,48,19,30)(2,25,8,31,14,37,20,43)(3,38,9,44,15,26,21,32)(4,27,10,33,16,39,22,45)(5,40,11,46,17,28,23,34)(6,29,12,35,18,41,24,47), (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)(25,43,37,31)(26,36,38,48)(27,29,39,41)(28,46,40,34)(30,32,42,44)(33,35,45,47)>;
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)(25,33,41)(26,42,34)(27,35,43)(28,44,36)(29,37,45)(30,46,38)(31,39,47)(32,48,40), (1,36,7,42,13,48,19,30)(2,25,8,31,14,37,20,43)(3,38,9,44,15,26,21,32)(4,27,10,33,16,39,22,45)(5,40,11,46,17,28,23,34)(6,29,12,35,18,41,24,47), (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)(25,43,37,31)(26,36,38,48)(27,29,39,41)(28,46,40,34)(30,32,42,44)(33,35,45,47) );
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),(25,33,41),(26,42,34),(27,35,43),(28,44,36),(29,37,45),(30,46,38),(31,39,47),(32,48,40)], [(1,36,7,42,13,48,19,30),(2,25,8,31,14,37,20,43),(3,38,9,44,15,26,21,32),(4,27,10,33,16,39,22,45),(5,40,11,46,17,28,23,34),(6,29,12,35,18,41,24,47)], [(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),(25,43,37,31),(26,36,38,48),(27,29,39,41),(28,46,40,34),(30,32,42,44),(33,35,45,47)])
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