metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊20D4, C6.412+ 1+4, C4⋊C4⋊24D6, (C2×D4)⋊8D6, D6⋊9(C4○D4), C4⋊D4⋊15S3, C3⋊6(D4⋊5D4), C22⋊C4⋊28D6, D6.19(C2×D4), C4.110(S3×D4), (C22×C4)⋊22D6, D6⋊3D4⋊21C2, C23⋊2D6⋊11C2, D6⋊C4⋊18C22, C4.D12⋊22C2, C12.229(C2×D4), (C6×D4)⋊14C22, Dic3⋊5D4⋊21C2, C6.71(C22×D4), C23.9D6⋊20C2, (C2×C6).156C24, C4⋊Dic3⋊32C22, C2.43(D4⋊6D6), C23.12D6⋊17C2, (C2×C12).595C23, Dic3⋊C4⋊65C22, (C22×C12)⋊22C22, (C4×Dic3)⋊23C22, (C2×Dic6)⋊62C22, (C22×C6).23C23, (C2×D12).264C22, C6.D4⋊24C22, (S3×C23).48C22, C23.123(C22×S3), C22.177(S3×C23), (C2×Dic3).75C23, (C22×S3).190C23, (C2×S3×D4)⋊13C2, C2.44(C2×S3×D4), (C4×C3⋊D4)⋊18C2, (S3×C22⋊C4)⋊6C2, (S3×C2×C4)⋊15C22, C2.40(S3×C4○D4), (C2×C4○D12)⋊22C2, (C3×C4⋊D4)⋊18C2, (C3×C4⋊C4)⋊13C22, C6.153(C2×C4○D4), (C2×C3⋊D4)⋊40C22, (C2×C4).39(C22×S3), (C3×C22⋊C4)⋊15C22, SmallGroup(192,1171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊20D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a7, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 992 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, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, S3×D4, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, D4⋊5D4, S3×C22⋊C4, C23.9D6, Dic3⋊5D4, C4.D12, C4×C3⋊D4, C23.12D6, C23⋊2D6, D6⋊3D4, C3×C4⋊D4, C2×C4○D12, C2×S3×D4, D12⋊20D4
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, D4⋊6D6, S3×C4○D4, D12⋊20D4
►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 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)
(1 21 32 40)(2 16 33 47)(3 23 34 42)(4 18 35 37)(5 13 36 44)(6 20 25 39)(7 15 26 46)(8 22 27 41)(9 17 28 48)(10 24 29 43)(11 19 30 38)(12 14 31 45)
(1 32)(2 27)(3 34)(4 29)(5 36)(6 31)(7 26)(8 33)(9 28)(10 35)(11 30)(12 25)(14 20)(16 22)(18 24)(37 43)(39 45)(41 47)
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)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43), (1,21,32,40)(2,16,33,47)(3,23,34,42)(4,18,35,37)(5,13,36,44)(6,20,25,39)(7,15,26,46)(8,22,27,41)(9,17,28,48)(10,24,29,43)(11,19,30,38)(12,14,31,45), (1,32)(2,27)(3,34)(4,29)(5,36)(6,31)(7,26)(8,33)(9,28)(10,35)(11,30)(12,25)(14,20)(16,22)(18,24)(37,43)(39,45)(41,47)>;
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)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43), (1,21,32,40)(2,16,33,47)(3,23,34,42)(4,18,35,37)(5,13,36,44)(6,20,25,39)(7,15,26,46)(8,22,27,41)(9,17,28,48)(10,24,29,43)(11,19,30,38)(12,14,31,45), (1,32)(2,27)(3,34)(4,29)(5,36)(6,31)(7,26)(8,33)(9,28)(10,35)(11,30)(12,25)(14,20)(16,22)(18,24)(37,43)(39,45)(41,47) );
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),(25,26),(27,36),(28,35),(29,34),(30,33),(31,32),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43)], [(1,21,32,40),(2,16,33,47),(3,23,34,42),(4,18,35,37),(5,13,36,44),(6,20,25,39),(7,15,26,46),(8,22,27,41),(9,17,28,48),(10,24,29,43),(11,19,30,38),(12,14,31,45)], [(1,32),(2,27),(3,34),(4,29),(5,36),(6,31),(7,26),(8,33),(9,28),(10,35),(11,30),(12,25),(14,20),(16,22),(18,24),(37,43),(39,45),(41,47)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | ··· | 2L | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| 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 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 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 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | S3×D4 | D4⋊6D6 | S3×C4○D4 |
| kernel | D12⋊20D4 | S3×C22⋊C4 | C23.9D6 | Dic3⋊5D4 | C4.D12 | C4×C3⋊D4 | C23.12D6 | C23⋊2D6 | D6⋊3D4 | C3×C4⋊D4 | C2×C4○D12 | C2×S3×D4 | C4⋊D4 | D12 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 3 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12⋊20D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;
D12⋊20D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{20}D_4 % in TeX
G:=Group("D12:20D4"); // GroupNames label
G:=SmallGroup(192,1171);
// by ID
G=gap.SmallGroup(192,1171);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^7,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations