metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊4D6, SD16⋊2D6, M4(2)⋊8D6, C24.1C23, C12.20C24, Dic12⋊1C22, D12.13C23, Dic6.13C23, (C2×D4)⋊30D6, D8⋊S3⋊2C2, C8⋊C22⋊6S3, D8⋊3S3⋊1C2, (S3×C8)⋊2C22, D4⋊S3⋊6C22, C4○D4.44D6, D6.54(C2×D4), (C4×S3).43D4, C4.190(S3×D4), C8.D6⋊1C2, D4.D6⋊1C2, (S3×D4)⋊9C22, (C3×D8)⋊2C22, C3⋊C8.10C23, C8.1(C22×S3), C24⋊C2⋊2C22, C8⋊S3⋊2C22, Q8.7D6⋊1C2, Q8.14D6⋊9C2, C12.241(C2×D4), (C6×D4)⋊22C22, (S3×M4(2))⋊2C2, C4.20(S3×C23), D4.S3⋊5C22, (S3×Q8)⋊10C22, C3⋊Q16⋊3C22, C22.47(S3×D4), D12⋊6C22⋊10C2, C3⋊3(D8⋊C22), (C4×S3).30C23, Dic3.60(C2×D4), (C3×SD16)⋊2C22, (C22×S3).43D4, (C3×D4).13C23, D4.13(C22×S3), C6.121(C22×D4), (C3×Q8).13C23, Q8.23(C22×S3), D4⋊2S3⋊10C22, (C2×C12).111C23, (C2×Dic3).195D4, Q8⋊3S3⋊10C22, (C2×Dic6)⋊39C22, C4○D12.28C22, (C3×M4(2))⋊2C22, C4.Dic3⋊13C22, C2.94(C2×S3×D4), (S3×C4○D4)⋊4C2, (C3×C8⋊C22)⋊2C2, (C2×C6).66(C2×D4), (C2×D4⋊2S3)⋊26C2, (S3×C2×C4).161C22, (C2×C4).95(C22×S3), (C3×C4○D4).24C22, SmallGroup(192,1332)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊4D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, cac-1=a5, ad=da, cbc-1=dbd=a4b, dcd=c-1 >
Subgroups: 688 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C2×M4(2), C4○D8, C8⋊C22, C8⋊C22, C8.C22, C2×C4○D4, S3×C8, C8⋊S3, C24⋊C2, Dic12, C4.Dic3, D4⋊S3, D4.S3, C3⋊Q16, C3×M4(2), C3×D8, C3×SD16, C2×Dic6, S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, D4⋊2S3, S3×Q8, Q8⋊3S3, C22×Dic3, C2×C3⋊D4, C6×D4, C3×C4○D4, D8⋊C22, S3×M4(2), C8.D6, D8⋊S3, D8⋊3S3, D4.D6, Q8.7D6, D12⋊6C22, Q8.14D6, C3×C8⋊C22, C2×D4⋊2S3, S3×C4○D4, D8⋊4D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D8⋊C22, C2×S3×D4, D8⋊4D6
►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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 31)(26 30)(27 29)(33 35)(36 40)(37 39)(41 47)(42 46)(43 45)
(1 46 32 21 34 14)(2 43 25 18 35 11)(3 48 26 23 36 16)(4 45 27 20 37 13)(5 42 28 17 38 10)(6 47 29 22 39 15)(7 44 30 19 40 12)(8 41 31 24 33 9)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,31)(26,30)(27,29)(33,35)(36,40)(37,39)(41,47)(42,46)(43,45), (1,46,32,21,34,14)(2,43,25,18,35,11)(3,48,26,23,36,16)(4,45,27,20,37,13)(5,42,28,17,38,10)(6,47,29,22,39,15)(7,44,30,19,40,12)(8,41,31,24,33,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44)>;
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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,31)(26,30)(27,29)(33,35)(36,40)(37,39)(41,47)(42,46)(43,45), (1,46,32,21,34,14)(2,43,25,18,35,11)(3,48,26,23,36,16)(4,45,27,20,37,13)(5,42,28,17,38,10)(6,47,29,22,39,15)(7,44,30,19,40,12)(8,41,31,24,33,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44) );
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)], [(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(17,21),(18,20),(22,24),(25,31),(26,30),(27,29),(33,35),(36,40),(37,39),(41,47),(42,46),(43,45)], [(1,46,32,21,34,14),(2,43,25,18,35,11),(3,48,26,23,36,16),(4,45,27,20,37,13),(5,42,28,17,38,10),(6,47,29,22,39,15),(7,44,30,19,40,12),(8,41,31,24,33,9)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 12 | 12 | 12 | 2 | 4 | 8 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | S3×D4 | S3×D4 | D8⋊C22 | D8⋊4D6 |
| kernel | D8⋊4D6 | S3×M4(2) | C8.D6 | D8⋊S3 | D8⋊3S3 | D4.D6 | Q8.7D6 | D12⋊6C22 | Q8.14D6 | C3×C8⋊C22 | C2×D4⋊2S3 | S3×C4○D4 | C8⋊C22 | C4×S3 | C2×Dic3 | C22×S3 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 |
Matrix representation of D8⋊4D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 0 | 0 | 0 | 0 | 27 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,1,0,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,46,0] >;
D8⋊4D6 in GAP, Magma, Sage, TeX
D_8\rtimes_4D_6
% in TeX
G:=Group("D8:4D6"); // GroupNames label
G:=SmallGroup(192,1332);
// by ID
G=gap.SmallGroup(192,1332);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,1123,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations