p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊8SD16, C42.503C23, C4.242- 1+4, Q82⋊3C2, (C8×Q8)⋊22C2, C4⋊C4.276D4, Q8⋊8(C4○D4), Q8○2(Q8⋊C4), D4⋊2Q8⋊44C2, C4⋊SD16⋊44C2, (C4×SD16)⋊47C2, C2.62(D4○D8), C4.4D8⋊32C2, (C2×Q8).268D4, Q8⋊6D4.8C2, C4.48(C2×SD16), C4⋊C4.430C23, C4⋊C8.350C22, (C4×C8).280C22, (C2×C8).369C23, (C2×C4).554C24, C4⋊Q8.183C22, C2.62(Q8⋊5D4), (C4×D4).194C22, (C2×D4).267C23, C4⋊1D4.96C22, (C4×Q8).308C22, (C2×Q8).402C23, C2.32(C22×SD16), C4.Q8.175C22, C22.814(C22×D4), D4⋊C4.127C22, Q8⋊C4.216C22, (C2×SD16).170C22, C4.255(C2×C4○D4), (C2×C4).1100(C2×D4), SmallGroup(128,2094)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊8SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c3 >
Subgroups: 408 in 196 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C4×Q8, C4×Q8, C4⋊D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×SD16, C2×C4○D4, C4×SD16, C8×Q8, C4⋊SD16, D4⋊2Q8, C4.4D8, Q8⋊6D4, Q82, Q8⋊8SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, C22×SD16, D4○D8, Q8⋊8SD16
►On 64 points
(1 51 59 25)(2 52 60 26)(3 53 61 27)(4 54 62 28)(5 55 63 29)(6 56 64 30)(7 49 57 31)(8 50 58 32)(9 20 33 46)(10 21 34 47)(11 22 35 48)(12 23 36 41)(13 24 37 42)(14 17 38 43)(15 18 39 44)(16 19 40 45)
(1 46 59 20)(2 21 60 47)(3 48 61 22)(4 23 62 41)(5 42 63 24)(6 17 64 43)(7 44 57 18)(8 19 58 45)(9 51 33 25)(10 26 34 52)(11 53 35 27)(12 28 36 54)(13 55 37 29)(14 30 38 56)(15 49 39 31)(16 32 40 50)
(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 33)(2 36)(3 39)(4 34)(5 37)(6 40)(7 35)(8 38)(9 59)(10 62)(11 57)(12 60)(13 63)(14 58)(15 61)(16 64)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)(41 52)(42 55)(43 50)(44 53)(45 56)(46 51)(47 54)(48 49)
G:=sub<Sym(64)| (1,51,59,25)(2,52,60,26)(3,53,61,27)(4,54,62,28)(5,55,63,29)(6,56,64,30)(7,49,57,31)(8,50,58,32)(9,20,33,46)(10,21,34,47)(11,22,35,48)(12,23,36,41)(13,24,37,42)(14,17,38,43)(15,18,39,44)(16,19,40,45), (1,46,59,20)(2,21,60,47)(3,48,61,22)(4,23,62,41)(5,42,63,24)(6,17,64,43)(7,44,57,18)(8,19,58,45)(9,51,33,25)(10,26,34,52)(11,53,35,27)(12,28,36,54)(13,55,37,29)(14,30,38,56)(15,49,39,31)(16,32,40,50), (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), (1,33)(2,36)(3,39)(4,34)(5,37)(6,40)(7,35)(8,38)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(41,52)(42,55)(43,50)(44,53)(45,56)(46,51)(47,54)(48,49)>;
G:=Group( (1,51,59,25)(2,52,60,26)(3,53,61,27)(4,54,62,28)(5,55,63,29)(6,56,64,30)(7,49,57,31)(8,50,58,32)(9,20,33,46)(10,21,34,47)(11,22,35,48)(12,23,36,41)(13,24,37,42)(14,17,38,43)(15,18,39,44)(16,19,40,45), (1,46,59,20)(2,21,60,47)(3,48,61,22)(4,23,62,41)(5,42,63,24)(6,17,64,43)(7,44,57,18)(8,19,58,45)(9,51,33,25)(10,26,34,52)(11,53,35,27)(12,28,36,54)(13,55,37,29)(14,30,38,56)(15,49,39,31)(16,32,40,50), (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), (1,33)(2,36)(3,39)(4,34)(5,37)(6,40)(7,35)(8,38)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(41,52)(42,55)(43,50)(44,53)(45,56)(46,51)(47,54)(48,49) );
G=PermutationGroup([[(1,51,59,25),(2,52,60,26),(3,53,61,27),(4,54,62,28),(5,55,63,29),(6,56,64,30),(7,49,57,31),(8,50,58,32),(9,20,33,46),(10,21,34,47),(11,22,35,48),(12,23,36,41),(13,24,37,42),(14,17,38,43),(15,18,39,44),(16,19,40,45)], [(1,46,59,20),(2,21,60,47),(3,48,61,22),(4,23,62,41),(5,42,63,24),(6,17,64,43),(7,44,57,18),(8,19,58,45),(9,51,33,25),(10,26,34,52),(11,53,35,27),(12,28,36,54),(13,55,37,29),(14,30,38,56),(15,49,39,31),(16,32,40,50)], [(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)], [(1,33),(2,36),(3,39),(4,34),(5,37),(6,40),(7,35),(8,38),(9,59),(10,62),(11,57),(12,60),(13,63),(14,58),(15,61),(16,64),(17,32),(18,27),(19,30),(20,25),(21,28),(22,31),(23,26),(24,29),(41,52),(42,55),(43,50),(44,53),(45,56),(46,51),(47,54),(48,49)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | ··· | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 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 | D4 | SD16 | C4○D4 | 2- 1+4 | D4○D8 |
| kernel | Q8⋊8SD16 | C4×SD16 | C8×Q8 | C4⋊SD16 | D4⋊2Q8 | C4.4D8 | Q8⋊6D4 | Q82 | C4⋊C4 | C2×Q8 | Q8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 8 | 4 | 1 | 2 |
Matrix representation of Q8⋊8SD16 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 12 | 5 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,13],[12,12,0,0,5,12,0,0,0,0,0,13,0,0,4,0],[0,16,0,0,16,0,0,0,0,0,0,4,0,0,13,0] >;
Q8⋊8SD16 in GAP, Magma, Sage, TeX
Q_8\rtimes_8{\rm SD}_{16} % in TeX
G:=Group("Q8:8SD16"); // GroupNames label
G:=SmallGroup(128,2094);
// by ID
G=gap.SmallGroup(128,2094);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations