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, (C3×D8)⋊2C22, (S3×D4)⋊9C22, C3⋊C8.10C23, C8.1(C22×S3), C24⋊C2⋊2C22, C8⋊S3⋊2C22, Q8.14D6⋊9C2, Q8.7D6⋊1C2, 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
Subgroups: 688 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×6], C22, C22 [×11], S3 [×3], C6, C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4, D4 [×2], D4 [×11], Q8, Q8 [×5], C23 [×3], Dic3 [×2], Dic3 [×3], C12 [×2], C12, D6 [×2], D6 [×4], C2×C6, C2×C6 [×5], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×2], D8 [×2], SD16 [×2], SD16 [×6], Q16 [×4], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8 [×2], C4○D4, C4○D4 [×11], C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×2], Dic6 [×2], C4×S3 [×4], C4×S3 [×3], D12, D12, C2×Dic3, C2×Dic3 [×6], C3⋊D4 [×7], C2×C12, C2×C12, C3×D4, C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×S3, C22×S3, C22×C6, C2×M4(2), C4○D8 [×4], C8⋊C22, C8⋊C22 [×3], C8.C22 [×4], C2×C4○D4 [×2], S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], Dic12 [×2], C4.Dic3, D4⋊S3 [×2], D4.S3 [×4], C3⋊Q16 [×2], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C2×Dic6, S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3 [×4], D4⋊2S3 [×3], S3×Q8, Q8⋊3S3, C22×Dic3, C2×C3⋊D4, C6×D4, C3×C4○D4, D8⋊C22, S3×M4(2), C8.D6, D8⋊S3 [×2], D8⋊3S3 [×2], D4.D6 [×2], Q8.7D6 [×2], D12⋊6C22, Q8.14D6, C3×C8⋊C22, C2×D4⋊2S3, S3×C4○D4, D8⋊4D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, S3×D4 [×2], S3×C23, D8⋊C22, C2×S3×D4, D8⋊4D6
Generators and relations
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 >
►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 23)(18 22)(19 21)(25 31)(26 30)(27 29)(33 39)(34 38)(35 37)(41 47)(42 46)(43 45)
(1 46 32 18 40 14)(2 43 25 23 33 11)(3 48 26 20 34 16)(4 45 27 17 35 13)(5 42 28 22 36 10)(6 47 29 19 37 15)(7 44 30 24 38 12)(8 41 31 21 39 9)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
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)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(33,39)(34,38)(35,37)(41,47)(42,46)(43,45), (1,46,32,18,40,14)(2,43,25,23,33,11)(3,48,26,20,34,16)(4,45,27,17,35,13)(5,42,28,22,36,10)(6,47,29,19,37,15)(7,44,30,24,38,12)(8,41,31,21,39,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)>;
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)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(33,39)(34,38)(35,37)(41,47)(42,46)(43,45), (1,46,32,18,40,14)(2,43,25,23,33,11)(3,48,26,20,34,16)(4,45,27,17,35,13)(5,42,28,22,36,10)(6,47,29,19,37,15)(7,44,30,24,38,12)(8,41,31,21,39,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46) );
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),(17,23),(18,22),(19,21),(25,31),(26,30),(27,29),(33,39),(34,38),(35,37),(41,47),(42,46),(43,45)], [(1,46,32,18,40,14),(2,43,25,23,33,11),(3,48,26,20,34,16),(4,45,27,17,35,13),(5,42,28,22,36,10),(6,47,29,19,37,15),(7,44,30,24,38,12),(8,41,31,21,39,9)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)])
Matrix representation ►G ⊆ 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] >;
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 |
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