p-group, metabelian, nilpotent (class 4), monomial
Aliases: D8⋊3C8, C42.33D4, C8.1M4(2), C4.3C4≀C2, C8.1(C2×C8), C8⋊2C8⋊6C2, C16⋊5C4⋊7C2, C2.8(D4⋊C8), (C4×D8).14C2, (C2×D8).10C4, (C2×C4).100D8, (C2×C8).297D4, C2.D8.15C4, C4.3(C22⋊C8), (C2×C4).84SD16, C2.1(D8⋊2C4), (C4×C8).129C22, C2.1(M5(2)⋊C2), C22.42(D4⋊C4), (C2×C8).46(C2×C4), (C2×C4).213(C22⋊C4), SmallGroup(128,65)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊C8
G = < a,b,c | a8=b2=c8=1, bab=a-1, cac-1=a3, cbc-1=a5b >
Smallest permutation representation of D8⋊C8
►On 64 points
Generators in S64
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 32)(8 31)(9 35)(10 34)(11 33)(12 40)(13 39)(14 38)(15 37)(16 36)(17 42)(18 41)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(49 60)(50 59)(51 58)(52 57)(53 64)(54 63)(55 62)(56 61)
(1 19 35 61 31 42 14 54)(2 22 36 64 32 45 15 49)(3 17 37 59 25 48 16 52)(4 20 38 62 26 43 9 55)(5 23 39 57 27 46 10 50)(6 18 40 60 28 41 11 53)(7 21 33 63 29 44 12 56)(8 24 34 58 30 47 13 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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,32)(8,31)(9,35)(10,34)(11,33)(12,40)(13,39)(14,38)(15,37)(16,36)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(49,60)(50,59)(51,58)(52,57)(53,64)(54,63)(55,62)(56,61), (1,19,35,61,31,42,14,54)(2,22,36,64,32,45,15,49)(3,17,37,59,25,48,16,52)(4,20,38,62,26,43,9,55)(5,23,39,57,27,46,10,50)(6,18,40,60,28,41,11,53)(7,21,33,63,29,44,12,56)(8,24,34,58,30,47,13,51)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,32)(8,31)(9,35)(10,34)(11,33)(12,40)(13,39)(14,38)(15,37)(16,36)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(49,60)(50,59)(51,58)(52,57)(53,64)(54,63)(55,62)(56,61), (1,19,35,61,31,42,14,54)(2,22,36,64,32,45,15,49)(3,17,37,59,25,48,16,52)(4,20,38,62,26,43,9,55)(5,23,39,57,27,46,10,50)(6,18,40,60,28,41,11,53)(7,21,33,63,29,44,12,56)(8,24,34,58,30,47,13,51) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,32),(8,31),(9,35),(10,34),(11,33),(12,40),(13,39),(14,38),(15,37),(16,36),(17,42),(18,41),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(49,60),(50,59),(51,58),(52,57),(53,64),(54,63),(55,62),(56,61)], [(1,19,35,61,31,42,14,54),(2,22,36,64,32,45,15,49),(3,17,37,59,25,48,16,52),(4,20,38,62,26,43,9,55),(5,23,39,57,27,46,10,50),(6,18,40,60,28,41,11,53),(7,21,33,63,29,44,12,56),(8,24,34,58,30,47,13,51)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D4 | M4(2) | D8 | SD16 | C4≀C2 | D8⋊2C4 | M5(2)⋊C2 |
| kernel | D8⋊C8 | C8⋊2C8 | C16⋊5C4 | C4×D8 | C2.D8 | C2×D8 | D8 | C42 | C2×C8 | C8 | C2×C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of D8⋊C8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 5 | 7 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 10 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 14 | 4 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 12 | 4 | 5 |
| 0 | 0 | 3 | 3 | 15 | 13 |
| 0 | 0 | 4 | 5 | 14 | 12 |
| 0 | 0 | 15 | 13 | 3 | 3 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,0,0,7,7,0,0,0,5,0,0,0,0,7,7,0,0],[1,10,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,12,0,0,0,0,7,0,0,0],[14,1,0,0,0,0,4,3,0,0,0,0,0,0,14,3,4,15,0,0,12,3,5,13,0,0,4,15,14,3,0,0,5,13,12,3] >;
D8⋊C8 in GAP, Magma, Sage, TeX
D_8\rtimes C_8
% in TeX
G:=Group("D8:C8"); // GroupNames label
G:=SmallGroup(128,65);
// by ID
G=gap.SmallGroup(128,65);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,891,100,1018,136,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^5*b>;
// generators/relations
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