p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊6Q8, C42.521C23, C4.952- 1+4, C4⋊C4○3D8, (C8×Q8)⋊14C2, D4.9(C2×Q8), C2.38(D4×Q8), C8.35(C2×Q8), C4⋊C4.415D4, C8⋊2Q8⋊22C2, (C4×D8).10C2, D4.Q8⋊11C2, D4⋊3Q8⋊15C2, D4⋊2Q8⋊45C2, C2.66(D4○D8), C4.49(C4○D8), (C2×Q8).186D4, C8.5Q8⋊10C2, C4.38(C22×Q8), C4⋊C4.269C23, C4⋊C8.327C22, (C2×C8).211C23, (C4×C8).123C22, (C2×C4).572C24, C4⋊Q8.201C22, C2.D8.71C22, (C2×D8).177C22, (C4×D4).210C22, (C2×D4).435C23, (C4×Q8).311C22, C4.Q8.115C22, C22.832(C22×D4), C42.C2.71C22, D4⋊C4.190C22, C2.76(C2×C4○D8), (C2×C4).178(C2×D4), SmallGroup(128,2112)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊6Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 328 in 176 conjugacy classes, 96 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×D8, C4×D8, C4×D8, C8×Q8, D4⋊2Q8, D4.Q8, C8.5Q8, C8⋊2Q8, D4⋊3Q8, D8⋊6Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4○D8, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C4○D8, D4○D8, D8⋊6Q8
►On 64 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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 32)(7 31)(8 30)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(17 51)(18 50)(19 49)(20 56)(21 55)(22 54)(23 53)(24 52)(41 59)(42 58)(43 57)(44 64)(45 63)(46 62)(47 61)(48 60)
(1 35 29 13)(2 36 30 14)(3 37 31 15)(4 38 32 16)(5 39 25 9)(6 40 26 10)(7 33 27 11)(8 34 28 12)(17 42 49 64)(18 43 50 57)(19 44 51 58)(20 45 52 59)(21 46 53 60)(22 47 54 61)(23 48 55 62)(24 41 56 63)
(1 63 29 41)(2 64 30 42)(3 57 31 43)(4 58 32 44)(5 59 25 45)(6 60 26 46)(7 61 27 47)(8 62 28 48)(9 20 39 52)(10 21 40 53)(11 22 33 54)(12 23 34 55)(13 24 35 56)(14 17 36 49)(15 18 37 50)(16 19 38 51)
G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(17,51)(18,50)(19,49)(20,56)(21,55)(22,54)(23,53)(24,52)(41,59)(42,58)(43,57)(44,64)(45,63)(46,62)(47,61)(48,60), (1,35,29,13)(2,36,30,14)(3,37,31,15)(4,38,32,16)(5,39,25,9)(6,40,26,10)(7,33,27,11)(8,34,28,12)(17,42,49,64)(18,43,50,57)(19,44,51,58)(20,45,52,59)(21,46,53,60)(22,47,54,61)(23,48,55,62)(24,41,56,63), (1,63,29,41)(2,64,30,42)(3,57,31,43)(4,58,32,44)(5,59,25,45)(6,60,26,46)(7,61,27,47)(8,62,28,48)(9,20,39,52)(10,21,40,53)(11,22,33,54)(12,23,34,55)(13,24,35,56)(14,17,36,49)(15,18,37,50)(16,19,38,51)>;
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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(17,51)(18,50)(19,49)(20,56)(21,55)(22,54)(23,53)(24,52)(41,59)(42,58)(43,57)(44,64)(45,63)(46,62)(47,61)(48,60), (1,35,29,13)(2,36,30,14)(3,37,31,15)(4,38,32,16)(5,39,25,9)(6,40,26,10)(7,33,27,11)(8,34,28,12)(17,42,49,64)(18,43,50,57)(19,44,51,58)(20,45,52,59)(21,46,53,60)(22,47,54,61)(23,48,55,62)(24,41,56,63), (1,63,29,41)(2,64,30,42)(3,57,31,43)(4,58,32,44)(5,59,25,45)(6,60,26,46)(7,61,27,47)(8,62,28,48)(9,20,39,52)(10,21,40,53)(11,22,33,54)(12,23,34,55)(13,24,35,56)(14,17,36,49)(15,18,37,50)(16,19,38,51) );
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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,32),(7,31),(8,30),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,40),(17,51),(18,50),(19,49),(20,56),(21,55),(22,54),(23,53),(24,52),(41,59),(42,58),(43,57),(44,64),(45,63),(46,62),(47,61),(48,60)], [(1,35,29,13),(2,36,30,14),(3,37,31,15),(4,38,32,16),(5,39,25,9),(6,40,26,10),(7,33,27,11),(8,34,28,12),(17,42,49,64),(18,43,50,57),(19,44,51,58),(20,45,52,59),(21,46,53,60),(22,47,54,61),(23,48,55,62),(24,41,56,63)], [(1,63,29,41),(2,64,30,42),(3,57,31,43),(4,58,32,44),(5,59,25,45),(6,60,26,46),(7,61,27,47),(8,62,28,48),(9,20,39,52),(10,21,40,53),(11,22,33,54),(12,23,34,55),(13,24,35,56),(14,17,36,49),(15,18,37,50),(16,19,38,51)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C4○D8 | 2- 1+4 | D4○D8 |
| kernel | D8⋊6Q8 | C4×D8 | C8×Q8 | D4⋊2Q8 | D4.Q8 | C8.5Q8 | C8⋊2Q8 | D4⋊3Q8 | C4⋊C4 | D8 | C2×Q8 | C4 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 4 | 2 | 1 | 2 | 3 | 4 | 1 | 8 | 1 | 2 |
Matrix representation of D8⋊6Q8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 14 | 6 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
| 16 | 16 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 16 | 0 | 0 |
| 14 | 8 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 4 | 13 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,14,0,0,6,6],[16,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16],[16,2,0,0,16,1,0,0,0,0,1,0,0,0,0,1],[9,14,0,0,16,8,0,0,0,0,4,4,0,0,9,13] >;
D8⋊6Q8 in GAP, Magma, Sage, TeX
D_8\rtimes_6Q_8
% in TeX
G:=Group("D8:6Q8"); // GroupNames label
G:=SmallGroup(128,2112);
// by ID
G=gap.SmallGroup(128,2112);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations