metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊13D6, D12.28D4, C12.4C24, D24⋊16C22, C24.40C23, Dic6.28D4, D12.2C23, Dic6.2C23, Dic12⋊14C22, (C2×C8)⋊9D6, (C6×D8)⋊3C2, (S3×D8)⋊6C2, C3⋊2(D4○D8), C4○D24⋊3C2, C8○D12⋊2C2, (C2×D8)⋊12S3, (C2×D4)⋊14D6, D8⋊S3⋊5C2, C4.75(S3×D4), C3⋊D4.8D4, C3⋊C8.1C23, D8⋊3S3⋊6C2, D4⋊6D6⋊5C2, (S3×C8)⋊7C22, (C2×C24)⋊3C22, D4⋊S3⋊1C22, D6.26(C2×D4), C12.79(C2×D4), (S3×D4)⋊1C22, C4.4(S3×C23), D12⋊6C22⋊7C2, C4○D12⋊3C22, (C6×D4)⋊20C22, (C3×D8)⋊11C22, (C4×S3).2C23, C8.10(C22×S3), D4.S3⋊1C22, (C3×D4).2C23, D4.2(C22×S3), C22.20(S3×D4), C24⋊C2⋊14C22, C8⋊S3⋊13C22, D4⋊2S3⋊1C22, Dic3.31(C2×D4), C6.105(C22×D4), (C2×C12).521C23, C4.Dic3⋊28C22, C2.78(C2×S3×D4), (C2×C6).394(C2×D4), (C2×C4).229(C22×S3), SmallGroup(192,1316)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊13D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a6b, dbd=a4b, dcd=c-1 >
Subgroups: 808 in 268 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, D8, SD16, Q16, C2×D4, C2×D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C8○D4, C2×D8, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, S3×C8, C8⋊S3, C24⋊C2, D24, Dic12, C4.Dic3, D4⋊S3, D4.S3, C2×C24, C3×D8, C4○D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, C2×C3⋊D4, C6×D4, D4○D8, C8○D12, C4○D24, S3×D8, D8⋊S3, D8⋊3S3, D12⋊6C22, C6×D8, D4⋊6D6, D8⋊13D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D4○D8, C2×S3×D4, D8⋊13D6
►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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 33)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)
(1 12 46 2 11 47)(3 10 48 8 13 45)(4 9 41 7 14 44)(5 16 42 6 15 43)(17 38 31 22 33 28)(18 37 32 21 34 27)(19 36 25 20 35 26)(23 40 29 24 39 30)
(1 46)(2 47)(3 48)(4 41)(5 42)(6 43)(7 44)(8 45)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 37)(34 38)(35 39)(36 40)
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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,33)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43), (1,12,46,2,11,47)(3,10,48,8,13,45)(4,9,41,7,14,44)(5,16,42,6,15,43)(17,38,31,22,33,28)(18,37,32,21,34,27)(19,36,25,20,35,26)(23,40,29,24,39,30), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,37)(34,38)(35,39)(36,40)>;
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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,33)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43), (1,12,46,2,11,47)(3,10,48,8,13,45)(4,9,41,7,14,44)(5,16,42,6,15,43)(17,38,31,22,33,28)(18,37,32,21,34,27)(19,36,25,20,35,26)(23,40,29,24,39,30), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,37)(34,38)(35,39)(36,40) );
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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,33),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(25,42),(26,41),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43)], [(1,12,46,2,11,47),(3,10,48,8,13,45),(4,9,41,7,14,44),(5,16,42,6,15,43),(17,38,31,22,33,28),(18,37,32,21,34,27),(19,36,25,20,35,26),(23,40,29,24,39,30)], [(1,46),(2,47),(3,48),(4,41),(5,42),(6,43),(7,44),(8,45),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,37),(34,38),(35,39),(36,40)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 4 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 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 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | S3×D4 | S3×D4 | D4○D8 | D8⋊13D6 |
| kernel | D8⋊13D6 | C8○D12 | C4○D24 | S3×D8 | D8⋊S3 | D8⋊3S3 | D12⋊6C22 | C6×D8 | D4⋊6D6 | C2×D8 | Dic6 | D12 | C3⋊D4 | C2×C8 | D8 | C2×D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 1 | 1 | 2 | 4 |
Matrix representation of D8⋊13D6 ►in GL4(𝔽7) generated by
| 3 | 5 | 2 | 2 |
| 4 | 5 | 1 | 2 |
| 6 | 6 | 5 | 2 |
| 1 | 6 | 3 | 0 |
| 3 | 2 | 6 | 3 |
| 6 | 0 | 4 | 5 |
| 6 | 6 | 5 | 2 |
| 0 | 0 | 0 | 6 |
| 0 | 4 | 6 | 0 |
| 2 | 5 | 2 | 0 |
| 1 | 6 | 3 | 5 |
| 3 | 6 | 3 | 6 |
| 1 | 0 | 2 | 4 |
| 2 | 2 | 5 | 5 |
| 6 | 3 | 1 | 6 |
| 4 | 2 | 6 | 3 |
G:=sub<GL(4,GF(7))| [3,4,6,1,5,5,6,6,2,1,5,3,2,2,2,0],[3,6,6,0,2,0,6,0,6,4,5,0,3,5,2,6],[0,2,1,3,4,5,6,6,6,2,3,3,0,0,5,6],[1,2,6,4,0,2,3,2,2,5,1,6,4,5,6,3] >;
D8⋊13D6 in GAP, Magma, Sage, TeX
D_8\rtimes_{13}D_6 % in TeX
G:=Group("D8:13D6"); // GroupNames label
G:=SmallGroup(192,1316);
// by ID
G=gap.SmallGroup(192,1316);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^6*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations