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⋊2Q8⋊45C2, D4⋊3Q8⋊15C2, C4.49(C4○D8), C2.66(D4○D8), (C2×Q8).186D4, C8.5Q8⋊10C2, C4.38(C22×Q8), C4⋊C8.327C22, C4⋊C4.269C23, (C2×C4).572C24, (C4×C8).123C22, (C2×C8).211C23, C4⋊Q8.201C22, C2.D8.71C22, (C2×D4).435C23, (C4×D4).210C22, (C2×D8).177C22, (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
Subgroups: 328 in 176 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×8], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], D4 [×2], Q8 [×4], C23 [×2], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×8], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×6], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C4×C8, C4×C8 [×2], D4⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×4], C2.D8, C2.D8 [×4], C2×C4⋊C4 [×2], C4×D4 [×6], C4×Q8, C22⋊Q8 [×6], C42.C2 [×4], C4⋊Q8 [×2], C2×D8, C4×D8, C4×D8 [×2], C8×Q8, D4⋊2Q8 [×2], D4.Q8 [×4], C8.5Q8 [×2], C8⋊2Q8, D4⋊3Q8 [×2], D8⋊6Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C4○D8 [×2], C22×D4, C22×Q8, 2- (1+4), D4×Q8, C2×C4○D8, D4○D8, D8⋊6Q8
Generators and relations
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 >
►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 49)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(41 61)(42 60)(43 59)(44 58)(45 57)(46 64)(47 63)(48 62)
(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 44 55 64)(18 45 56 57)(19 46 49 58)(20 47 50 59)(21 48 51 60)(22 41 52 61)(23 42 53 62)(24 43 54 63)
(1 63 29 43)(2 64 30 44)(3 57 31 45)(4 58 32 46)(5 59 25 47)(6 60 26 48)(7 61 27 41)(8 62 28 42)(9 20 39 50)(10 21 40 51)(11 22 33 52)(12 23 34 53)(13 24 35 54)(14 17 36 55)(15 18 37 56)(16 19 38 49)
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)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(41,61)(42,60)(43,59)(44,58)(45,57)(46,64)(47,63)(48,62), (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)(17,44,55,64)(18,45,56,57)(19,46,49,58)(20,47,50,59)(21,48,51,60)(22,41,52,61)(23,42,53,62)(24,43,54,63), (1,63,29,43)(2,64,30,44)(3,57,31,45)(4,58,32,46)(5,59,25,47)(6,60,26,48)(7,61,27,41)(8,62,28,42)(9,20,39,50)(10,21,40,51)(11,22,33,52)(12,23,34,53)(13,24,35,54)(14,17,36,55)(15,18,37,56)(16,19,38,49)>;
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)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(41,61)(42,60)(43,59)(44,58)(45,57)(46,64)(47,63)(48,62), (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)(17,44,55,64)(18,45,56,57)(19,46,49,58)(20,47,50,59)(21,48,51,60)(22,41,52,61)(23,42,53,62)(24,43,54,63), (1,63,29,43)(2,64,30,44)(3,57,31,45)(4,58,32,46)(5,59,25,47)(6,60,26,48)(7,61,27,41)(8,62,28,42)(9,20,39,50)(10,21,40,51)(11,22,33,52)(12,23,34,53)(13,24,35,54)(14,17,36,55)(15,18,37,56)(16,19,38,49) );
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),(17,49),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(41,61),(42,60),(43,59),(44,58),(45,57),(46,64),(47,63),(48,62)], [(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),(17,44,55,64),(18,45,56,57),(19,46,49,58),(20,47,50,59),(21,48,51,60),(22,41,52,61),(23,42,53,62),(24,43,54,63)], [(1,63,29,43),(2,64,30,44),(3,57,31,45),(4,58,32,46),(5,59,25,47),(6,60,26,48),(7,61,27,41),(8,62,28,42),(9,20,39,50),(10,21,40,51),(11,22,33,52),(12,23,34,53),(13,24,35,54),(14,17,36,55),(15,18,37,56),(16,19,38,49)])
Matrix representation ►G ⊆ 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] >;
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 |
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