metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.31D4, C23.41D12, (C2×C8)⋊15D6, C22⋊C8⋊8S3, (C2×C6)⋊1SD16, C6.7C22≀C2, (C2×C12).42D4, C2.D24⋊8C2, C4.119(S3×D4), (C2×C4).31D12, C6.7(C2×SD16), (C2×C24)⋊14C22, C12.331(C2×D4), C3⋊1(C22⋊SD16), C6.8(C8⋊C22), C4⋊Dic3⋊1C22, (C22×C4).97D6, (C22×C6).51D4, C22⋊4(C24⋊C2), C12.48D4⋊1C2, C2.11(C8⋊D6), (C2×Dic6)⋊1C22, (C22×D12).3C2, C2.10(D6⋊D4), (C2×C12).741C23, C22.104(C2×D12), (C2×D12).191C22, (C22×C12).50C22, (C2×C24⋊C2)⋊9C2, (C3×C22⋊C8)⋊10C2, C2.10(C2×C24⋊C2), (C2×C6).124(C2×D4), (C2×C4).686(C22×S3), SmallGroup(192,290)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.31D4
G = < a,b,c,d | a12=b2=d2=1, c4=a6, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=a3c3 >
Subgroups: 736 in 188 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C24, Dic6, D12, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C24⋊C2, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×C24, C2×Dic6, C2×D12, C2×D12, C22×C12, S3×C23, C22⋊SD16, C2.D24, C3×C22⋊C8, C2×C24⋊C2, C12.48D4, C22×D12, D12.31D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C22×S3, C22≀C2, C2×SD16, C8⋊C22, C24⋊C2, C2×D12, S3×D4, C22⋊SD16, D6⋊D4, C2×C24⋊C2, C8⋊D6, D12.31D4
►On 48 points
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 33)(26 32)(27 31)(28 30)(34 36)(38 48)(39 47)(40 46)(41 45)(42 44)
(1 31 15 48 7 25 21 42)(2 32 16 37 8 26 22 43)(3 33 17 38 9 27 23 44)(4 34 18 39 10 28 24 45)(5 35 19 40 11 29 13 46)(6 36 20 41 12 30 14 47)
(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)
G:=sub<Sym(48)| (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,33)(26,32)(27,31)(28,30)(34,36)(38,48)(39,47)(40,46)(41,45)(42,44), (1,31,15,48,7,25,21,42)(2,32,16,37,8,26,22,43)(3,33,17,38,9,27,23,44)(4,34,18,39,10,28,24,45)(5,35,19,40,11,29,13,46)(6,36,20,41,12,30,14,47), (25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,33)(26,32)(27,31)(28,30)(34,36)(38,48)(39,47)(40,46)(41,45)(42,44), (1,31,15,48,7,25,21,42)(2,32,16,37,8,26,22,43)(3,33,17,38,9,27,23,44)(4,34,18,39,10,28,24,45)(5,35,19,40,11,29,13,46)(6,36,20,41,12,30,14,47), (25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44) );
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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(25,33),(26,32),(27,31),(28,30),(34,36),(38,48),(39,47),(40,46),(41,45),(42,44)], [(1,31,15,48,7,25,21,42),(2,32,16,37,8,26,22,43),(3,33,17,38,9,27,23,44),(4,34,18,39,10,28,24,45),(5,35,19,40,11,29,13,46),(6,36,20,41,12,30,14,47)], [(25,45),(26,46),(27,47),(28,48),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | SD16 | D12 | D12 | C24⋊C2 | C8⋊C22 | S3×D4 | C8⋊D6 |
| kernel | D12.31D4 | C2.D24 | C3×C22⋊C8 | C2×C24⋊C2 | C12.48D4 | C22×D12 | C22⋊C8 | D12 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 8 | 1 | 2 | 2 |
Matrix representation of D12.31D4 ►in GL4(𝔽73) generated by
| 66 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 66 | 7 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 12 | 1 |
| 48 | 11 | 0 | 0 |
| 62 | 37 | 0 | 0 |
| 0 | 0 | 61 | 71 |
| 0 | 0 | 35 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 61 | 72 |
G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,72,0,0,0,0,72],[66,14,0,0,7,7,0,0,0,0,72,12,0,0,0,1],[48,62,0,0,11,37,0,0,0,0,61,35,0,0,71,12],[1,0,0,0,0,1,0,0,0,0,1,61,0,0,0,72] >;
D12.31D4 in GAP, Magma, Sage, TeX
D_{12}._{31}D_4 % in TeX
G:=Group("D12.31D4"); // GroupNames label
G:=SmallGroup(192,290);
// by ID
G=gap.SmallGroup(192,290);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=d^2=1,c^4=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=a^3*c^3>;
// generators/relations