metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.11D12, Q8.16D12, D24⋊11C22, M4(2)⋊20D6, C24.10C23, C12.61C24, D12.24C23, Dic12⋊10C22, Dic6.24C23, (C2×C8)⋊6D6, C8○D4⋊7S3, D4○D12⋊3C2, Q8○D12⋊3C2, C4○D24⋊11C2, C8⋊D6⋊11C2, (C2×C24)⋊9C22, C4○D4.54D6, (C3×D4).23D4, C4.27(C2×D12), C3⋊1(D4○SD16), C12.73(C2×D4), (C3×Q8).23D4, C8.D6⋊11C2, C4○D12⋊1C22, C4.58(S3×C23), C8.55(C22×S3), C22.3(C2×D12), C6.28(C22×D4), C24⋊C2⋊11C22, C2.30(C22×D12), (C2×C12).515C23, (C2×Dic6)⋊35C22, (C2×D12).175C22, (C3×M4(2))⋊22C22, (C3×C8○D4)⋊3C2, (C2×C6).8(C2×D4), (C2×C24⋊C2)⋊6C2, (C2×C4).226(C22×S3), (C3×C4○D4).45C22, SmallGroup(192,1310)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.11D12
G = < a,b,c,d | a4=b2=1, c12=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c11 >
Subgroups: 752 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C24, Dic6, Dic6, Dic6, C4×S3, D12, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C24⋊C2, C24⋊C2, D24, Dic12, C2×C24, C3×M4(2), C2×Dic6, C2×D12, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, D4○SD16, C2×C24⋊C2, C4○D24, C8⋊D6, C8.D6, C3×C8○D4, D4○D12, Q8○D12, D4.11D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, C2×D12, S3×C23, D4○SD16, C22×D12, D4.11D12
►On 48 points
(1 25 13 37)(2 26 14 38)(3 27 15 39)(4 28 16 40)(5 29 17 41)(6 30 18 42)(7 31 19 43)(8 32 20 44)(9 33 21 45)(10 34 22 46)(11 35 23 47)(12 36 24 48)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)
(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 13 24)(2 23 14 11)(3 10 15 22)(4 21 16 9)(5 8 17 20)(6 19 18 7)(25 36 37 48)(26 47 38 35)(27 34 39 46)(28 45 40 33)(29 32 41 44)(30 43 42 31)
G:=sub<Sym(48)| (1,25,13,37)(2,26,14,38)(3,27,15,39)(4,28,16,40)(5,29,17,41)(6,30,18,42)(7,31,19,43)(8,32,20,44)(9,33,21,45)(10,34,22,46)(11,35,23,47)(12,36,24,48), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36), (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), (1,12,13,24)(2,23,14,11)(3,10,15,22)(4,21,16,9)(5,8,17,20)(6,19,18,7)(25,36,37,48)(26,47,38,35)(27,34,39,46)(28,45,40,33)(29,32,41,44)(30,43,42,31)>;
G:=Group( (1,25,13,37)(2,26,14,38)(3,27,15,39)(4,28,16,40)(5,29,17,41)(6,30,18,42)(7,31,19,43)(8,32,20,44)(9,33,21,45)(10,34,22,46)(11,35,23,47)(12,36,24,48), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36), (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), (1,12,13,24)(2,23,14,11)(3,10,15,22)(4,21,16,9)(5,8,17,20)(6,19,18,7)(25,36,37,48)(26,47,38,35)(27,34,39,46)(28,45,40,33)(29,32,41,44)(30,43,42,31) );
G=PermutationGroup([[(1,25,13,37),(2,26,14,38),(3,27,15,39),(4,28,16,40),(5,29,17,41),(6,30,18,42),(7,31,19,43),(8,32,20,44),(9,33,21,45),(10,34,22,46),(11,35,23,47),(12,36,24,48)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36)], [(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)], [(1,12,13,24),(2,23,14,11),(3,10,15,22),(4,21,16,9),(5,8,17,20),(6,19,18,7),(25,36,37,48),(26,47,38,35),(27,34,39,46),(28,45,40,33),(29,32,41,44),(30,43,42,31)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | D12 | D4○SD16 | D4.11D12 |
| kernel | D4.11D12 | C2×C24⋊C2 | C4○D24 | C8⋊D6 | C8.D6 | C3×C8○D4 | D4○D12 | Q8○D12 | C8○D4 | C3×D4 | C3×Q8 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 1 | 3 | 3 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of D4.11D12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 63 | 11 | 22 | 72 |
| 0 | 0 | 30 | 44 | 48 | 72 |
| 0 | 0 | 70 | 3 | 40 | 0 |
| 0 | 0 | 0 | 2 | 20 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 62 | 3 | 0 |
| 0 | 0 | 43 | 29 | 25 | 0 |
| 0 | 0 | 3 | 70 | 33 | 0 |
| 0 | 0 | 0 | 71 | 53 | 1 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 6 | 9 | 7 |
| 0 | 0 | 67 | 67 | 35 | 34 |
| 0 | 0 | 0 | 0 | 0 | 55 |
| 0 | 0 | 0 | 0 | 4 | 61 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 6 | 38 | 66 |
| 0 | 0 | 6 | 6 | 12 | 39 |
| 0 | 0 | 0 | 0 | 61 | 18 |
| 0 | 0 | 0 | 0 | 69 | 12 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,63,30,70,0,0,0,11,44,3,2,0,0,22,48,40,20,0,0,72,72,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,10,43,3,0,0,0,62,29,70,71,0,0,3,25,33,53,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,9,35,0,4,0,0,7,34,55,61],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,67,6,0,0,0,0,6,6,0,0,0,0,38,12,61,69,0,0,66,39,18,12] >;
D4.11D12 in GAP, Magma, Sage, TeX
D_4._{11}D_{12} % in TeX
G:=Group("D4.11D12"); // GroupNames label
G:=SmallGroup(192,1310);
// by ID
G=gap.SmallGroup(192,1310);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^12=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^11>;
// generators/relations