metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊11D4, C42⋊29D6, C6.772+ 1+4, C3⋊4D42, C4⋊2(S3×D4), D6⋊8(C2×D4), C12⋊3(C2×D4), (C2×D4)⋊26D6, C4⋊1D4⋊7S3, (C4×D12)⋊49C2, C23⋊2D6⋊27C2, D6⋊3D4⋊36C2, (C4×C12)⋊27C22, D6⋊C4⋊70C22, (C6×D4)⋊33C22, C6.95(C22×D4), (C2×C6).261C24, C4⋊Dic3⋊74C22, C2.81(D4⋊6D6), (C2×C12).509C23, (S3×C23)⋊13C22, (C22×C6).75C23, C23.77(C22×S3), (C2×D12).269C22, C6.D4⋊37C22, C22.282(S3×C23), (C22×S3).229C23, (C2×Dic3).136C23, (C2×S3×D4)⋊20C2, C2.68(C2×S3×D4), (C3×C4⋊1D4)⋊8C2, (S3×C2×C4)⋊29C22, (C2×C3⋊D4)⋊27C22, (C2×C4).214(C22×S3), SmallGroup(192,1276)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊11D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, dad=a7, cbc-1=a6b, bd=db, dcd=c-1 >
Subgroups: 1376 in 428 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, D42, C4×D12, C23⋊2D6, D6⋊3D4, C3×C4⋊1D4, C2×S3×D4, D12⋊11D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, D42, C2×S3×D4, D4⋊6D6, D12⋊11D4
►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 38)(2 37)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 25 39 22)(2 26 40 23)(3 27 41 24)(4 28 42 13)(5 29 43 14)(6 30 44 15)(7 31 45 16)(8 32 46 17)(9 33 47 18)(10 34 48 19)(11 35 37 20)(12 36 38 21)
(1 48)(2 43)(3 38)(4 45)(5 40)(6 47)(7 42)(8 37)(9 44)(10 39)(11 46)(12 41)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)(25 34)(26 29)(27 36)(28 31)(30 33)(32 35)
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), (1,38)(2,37)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28), (1,25,39,22)(2,26,40,23)(3,27,41,24)(4,28,42,13)(5,29,43,14)(6,30,44,15)(7,31,45,16)(8,32,46,17)(9,33,47,18)(10,34,48,19)(11,35,37,20)(12,36,38,21), (1,48)(2,43)(3,38)(4,45)(5,40)(6,47)(7,42)(8,37)(9,44)(10,39)(11,46)(12,41)(13,16)(14,23)(15,18)(17,20)(19,22)(21,24)(25,34)(26,29)(27,36)(28,31)(30,33)(32,35)>;
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), (1,38)(2,37)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28), (1,25,39,22)(2,26,40,23)(3,27,41,24)(4,28,42,13)(5,29,43,14)(6,30,44,15)(7,31,45,16)(8,32,46,17)(9,33,47,18)(10,34,48,19)(11,35,37,20)(12,36,38,21), (1,48)(2,43)(3,38)(4,45)(5,40)(6,47)(7,42)(8,37)(9,44)(10,39)(11,46)(12,41)(13,16)(14,23)(15,18)(17,20)(19,22)(21,24)(25,34)(26,29)(27,36)(28,31)(30,33)(32,35) );
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)], [(1,38),(2,37),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,27),(14,26),(15,25),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28)], [(1,25,39,22),(2,26,40,23),(3,27,41,24),(4,28,42,13),(5,29,43,14),(6,30,44,15),(7,31,45,16),(8,32,46,17),(9,33,47,18),(10,34,48,19),(11,35,37,20),(12,36,38,21)], [(1,48),(2,43),(3,38),(4,45),(5,40),(6,47),(7,42),(8,37),(9,44),(10,39),(11,46),(12,41),(13,16),(14,23),(15,18),(17,20),(19,22),(21,24),(25,34),(26,29),(27,36),(28,31),(30,33),(32,35)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2O | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | 2+ 1+4 | S3×D4 | D4⋊6D6 |
| kernel | D12⋊11D4 | C4×D12 | C23⋊2D6 | D6⋊3D4 | C3×C4⋊1D4 | C2×S3×D4 | C4⋊1D4 | D12 | C42 | C2×D4 | C6 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 1 | 4 | 1 | 8 | 1 | 6 | 1 | 4 | 2 |
Matrix representation of D12⋊11D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 8 | 6 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 12 | 3 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 5 | 7 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 3 | 6 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,7,8,0,0,0,0,10,6],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,6,10,0,0,0,0,3,7],[12,8,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,5,0,0,0,0,3,7],[12,8,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,7,3,0,0,0,0,10,6] >;
D12⋊11D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{11}D_4 % in TeX
G:=Group("D12:11D4"); // GroupNames label
G:=SmallGroup(192,1276);
// by ID
G=gap.SmallGroup(192,1276);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations