metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊13D4, C22⋊3D24, C23.42D12, (C2×C8)⋊1D6, (C2×C6)⋊1D8, C6.5(C2×D8), (C2×D24)⋊2C2, C22⋊C8⋊3S3, C2.7(C2×D24), C12⋊7D4⋊1C2, C3⋊1(C22⋊D8), (C2×C24)⋊1C22, C6.8C22≀C2, C4.120(S3×D4), (C2×C4).32D12, C2.D24⋊4C2, (C2×C12).43D4, C12.332(C2×D4), (C22×D12)⋊2C2, (C2×D12)⋊1C22, C6.9(C8⋊C22), C4⋊Dic3⋊2C22, (C22×C4).98D6, (C22×C6).52D4, C2.12(C8⋊D6), C2.11(D6⋊D4), (C2×C12).742C23, C22.105(C2×D12), (C22×C12).51C22, (C3×C22⋊C8)⋊5C2, (C2×C6).125(C2×D4), (C2×C4).687(C22×S3), SmallGroup(192,291)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊13D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a7b, bd=db, dcd=c-1 >
Subgroups: 832 in 198 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C24, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, D24, C4⋊Dic3, D6⋊C4, C2×C24, C2×D12, C2×D12, C2×D12, C2×C3⋊D4, C22×C12, S3×C23, C22⋊D8, C2.D24, C3×C22⋊C8, C2×D24, C12⋊7D4, C22×D12, D12⋊13D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, C22≀C2, C2×D8, C8⋊C22, D24, C2×D12, S3×D4, C22⋊D8, D6⋊D4, C2×D24, C8⋊D6, D12⋊13D4
►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 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 33)(26 32)(27 31)(28 30)(34 36)(37 39)(40 48)(41 47)(42 46)(43 45)
(1 46 22 31)(2 45 23 30)(3 44 24 29)(4 43 13 28)(5 42 14 27)(6 41 15 26)(7 40 16 25)(8 39 17 36)(9 38 18 35)(10 37 19 34)(11 48 20 33)(12 47 21 32)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
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)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,33)(26,32)(27,31)(28,30)(34,36)(37,39)(40,48)(41,47)(42,46)(43,45), (1,46,22,31)(2,45,23,30)(3,44,24,29)(4,43,13,28)(5,42,14,27)(6,41,15,26)(7,40,16,25)(8,39,17,36)(9,38,18,35)(10,37,19,34)(11,48,20,33)(12,47,21,32), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)>;
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)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,33)(26,32)(27,31)(28,30)(34,36)(37,39)(40,48)(41,47)(42,46)(43,45), (1,46,22,31)(2,45,23,30)(3,44,24,29)(4,43,13,28)(5,42,14,27)(6,41,15,26)(7,40,16,25)(8,39,17,36)(9,38,18,35)(10,37,19,34)(11,48,20,33)(12,47,21,32), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48) );
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),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,33),(26,32),(27,31),(28,30),(34,36),(37,39),(40,48),(41,47),(42,46),(43,45)], [(1,46,22,31),(2,45,23,30),(3,44,24,29),(4,43,13,28),(5,42,14,27),(6,41,15,26),(7,40,16,25),(8,39,17,36),(9,38,18,35),(10,37,19,34),(11,48,20,33),(12,47,21,32)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 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 | 2 | 3 | 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 | 24 | 2 | 2 | 2 | 4 | 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 | D8 | D12 | D12 | D24 | C8⋊C22 | S3×D4 | C8⋊D6 |
| kernel | D12⋊13D4 | C2.D24 | C3×C22⋊C8 | C2×D24 | C12⋊7D4 | 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⋊13D4 ►in GL4(𝔽73) generated by
| 66 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 66 | 7 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 50 | 68 | 0 | 0 |
| 18 | 23 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,1,0,0,0,0,1],[66,14,0,0,7,7,0,0,0,0,1,0,0,0,0,1],[50,18,0,0,68,23,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,1] >;
D12⋊13D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{13}D_4 % in TeX
G:=Group("D12:13D4"); // GroupNames label
G:=SmallGroup(192,291);
// by ID
G=gap.SmallGroup(192,291);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,226,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations