metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.11D12, Q8.16D12, D24⋊11C22, M4(2)⋊20D6, C12.61C24, C24.10C23, 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, C8.55(C22×S3), C4.58(S3×C23), C6.28(C22×D4), C22.3(C2×D12), 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
Subgroups: 752 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], S3 [×4], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4 [×3], D4 [×13], Q8, Q8 [×7], C23 [×3], Dic3 [×4], C12, C12 [×3], D6 [×7], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×4], C4○D4, C4○D4 [×10], C24, C24 [×3], Dic6, Dic6 [×3], Dic6 [×3], C4×S3 [×6], D12, D12 [×3], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C24⋊C2, C24⋊C2 [×9], D24 [×3], Dic12 [×3], C2×C24 [×3], C3×M4(2) [×3], C2×Dic6 [×3], C2×D12 [×3], C4○D12 [×6], S3×D4 [×3], D4⋊2S3 [×3], S3×Q8, Q8⋊3S3, C3×C4○D4, D4○SD16, C2×C24⋊C2 [×3], C4○D24 [×3], C8⋊D6 [×3], C8.D6 [×3], C3×C8○D4, D4○D12, Q8○D12, D4.11D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, D4○SD16, C22×D12, D4.11D12
Generators and relations
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 >
►On 48 points
(1 46 13 34)(2 47 14 35)(3 48 15 36)(4 25 16 37)(5 26 17 38)(6 27 18 39)(7 28 19 40)(8 29 20 41)(9 30 21 42)(10 31 22 43)(11 32 23 44)(12 33 24 45)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)
(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 42 37 30)(26 29 38 41)(27 40 39 28)(31 36 43 48)(32 47 44 35)(33 34 45 46)
G:=sub<Sym(48)| (1,46,13,34)(2,47,14,35)(3,48,15,36)(4,25,16,37)(5,26,17,38)(6,27,18,39)(7,28,19,40)(8,29,20,41)(9,30,21,42)(10,31,22,43)(11,32,23,44)(12,33,24,45), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33), (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)(25,42,37,30)(26,29,38,41)(27,40,39,28)(31,36,43,48)(32,47,44,35)(33,34,45,46)>;
G:=Group( (1,46,13,34)(2,47,14,35)(3,48,15,36)(4,25,16,37)(5,26,17,38)(6,27,18,39)(7,28,19,40)(8,29,20,41)(9,30,21,42)(10,31,22,43)(11,32,23,44)(12,33,24,45), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33), (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)(25,42,37,30)(26,29,38,41)(27,40,39,28)(31,36,43,48)(32,47,44,35)(33,34,45,46) );
G=PermutationGroup([(1,46,13,34),(2,47,14,35),(3,48,15,36),(4,25,16,37),(5,26,17,38),(6,27,18,39),(7,28,19,40),(8,29,20,41),(9,30,21,42),(10,31,22,43),(11,32,23,44),(12,33,24,45)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,25),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33)], [(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),(25,42,37,30),(26,29,38,41),(27,40,39,28),(31,36,43,48),(32,47,44,35),(33,34,45,46)])
Matrix representation ►G ⊆ 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] >;
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 |
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