metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊10D4, C42⋊21D6, C6.1252+ 1+4, (C2×Q8)⋊22D6, C4.70(S3×D4), (C4×D12)⋊43C2, C3⋊9(D4⋊5D4), C22⋊C4⋊33D6, D6.24(C2×D4), C12.63(C2×D4), D6⋊11(C4○D4), D6⋊D4⋊24C2, Dic3⋊D4⋊40C2, D6⋊3D4⋊33C2, (C4×C12)⋊23C22, D6⋊C4⋊54C22, D6⋊3Q8⋊28C2, (C2×D4).173D6, C4.4D4⋊10S3, (C6×Q8)⋊13C22, C6.90(C22×D4), C23.9D6⋊42C2, C2.49(D4○D12), (C2×D12)⋊28C22, (C2×C6).220C24, C4⋊Dic3⋊60C22, (C2×C12).601C23, Dic3⋊C4⋊26C22, (C6×D4).155C22, C23.52(C22×S3), (C22×C6).50C23, C6.D4⋊33C22, (S3×C23).64C22, C22.241(S3×C23), (C22×S3).215C23, (C2×Dic3).115C23, (C2×S3×D4)⋊17C2, C2.63(C2×S3×D4), (S3×C2×C4)⋊26C22, C2.76(S3×C4○D4), (S3×C22⋊C4)⋊17C2, C6.187(C2×C4○D4), (C2×Q8⋊3S3)⋊11C2, (C3×C4.4D4)⋊12C2, (C2×C3⋊D4)⋊23C22, (C3×C22⋊C4)⋊29C22, (C2×C4).195(C22×S3), SmallGroup(192,1235)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊10D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >
Subgroups: 1024 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, S3×D4, Q8⋊3S3, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, D4⋊5D4, C4×D12, S3×C22⋊C4, D6⋊D4, C23.9D6, Dic3⋊D4, D6⋊3D4, D6⋊3Q8, C3×C4.4D4, C2×S3×D4, C2×Q8⋊3S3, D12⋊10D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, S3×D4, S3×C23, D4⋊5D4, C2×S3×D4, S3×C4○D4, D4○D12, D12⋊10D4
►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 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)
(1 21 27 40)(2 22 28 41)(3 23 29 42)(4 24 30 43)(5 13 31 44)(6 14 32 45)(7 15 33 46)(8 16 34 47)(9 17 35 48)(10 18 36 37)(11 19 25 38)(12 20 26 39)
(1 15)(2 20)(3 13)(4 18)(5 23)(6 16)(7 21)(8 14)(9 19)(10 24)(11 17)(12 22)(25 48)(26 41)(27 46)(28 39)(29 44)(30 37)(31 42)(32 47)(33 40)(34 45)(35 38)(36 43)
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)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43), (1,21,27,40)(2,22,28,41)(3,23,29,42)(4,24,30,43)(5,13,31,44)(6,14,32,45)(7,15,33,46)(8,16,34,47)(9,17,35,48)(10,18,36,37)(11,19,25,38)(12,20,26,39), (1,15)(2,20)(3,13)(4,18)(5,23)(6,16)(7,21)(8,14)(9,19)(10,24)(11,17)(12,22)(25,48)(26,41)(27,46)(28,39)(29,44)(30,37)(31,42)(32,47)(33,40)(34,45)(35,38)(36,43)>;
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)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43), (1,21,27,40)(2,22,28,41)(3,23,29,42)(4,24,30,43)(5,13,31,44)(6,14,32,45)(7,15,33,46)(8,16,34,47)(9,17,35,48)(10,18,36,37)(11,19,25,38)(12,20,26,39), (1,15)(2,20)(3,13)(4,18)(5,23)(6,16)(7,21)(8,14)(9,19)(10,24)(11,17)(12,22)(25,48)(26,41)(27,46)(28,39)(29,44)(30,37)(31,42)(32,47)(33,40)(34,45)(35,38)(36,43) );
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),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43)], [(1,21,27,40),(2,22,28,41),(3,23,29,42),(4,24,30,43),(5,13,31,44),(6,14,32,45),(7,15,33,46),(8,16,34,47),(9,17,35,48),(10,18,36,37),(11,19,25,38),(12,20,26,39)], [(1,15),(2,20),(3,13),(4,18),(5,23),(6,16),(7,21),(8,14),(9,19),(10,24),(11,17),(12,22),(25,48),(26,41),(27,46),(28,39),(29,44),(30,37),(31,42),(32,47),(33,40),(34,45),(35,38),(36,43)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 2L | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | ··· | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | S3×D4 | S3×C4○D4 | D4○D12 |
| kernel | D12⋊10D4 | C4×D12 | S3×C22⋊C4 | D6⋊D4 | C23.9D6 | Dic3⋊D4 | D6⋊3D4 | D6⋊3Q8 | C3×C4.4D4 | C2×S3×D4 | C2×Q8⋊3S3 | C4.4D4 | D12 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12⋊10D4 ►in GL6(𝔽13)
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 8 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 10 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,3,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,10,1] >;
D12⋊10D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{10}D_4 % in TeX
G:=Group("D12:10D4"); // GroupNames label
G:=SmallGroup(192,1235);
// by ID
G=gap.SmallGroup(192,1235);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,192,6278]);
// 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^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations