p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊7SD16, C42.500C23, C4.212- 1+4, (C8×Q8)⋊21C2, (D4×Q8).8C2, C4⋊C4.273D4, Q8⋊3Q8⋊2C2, Q8○3(D4⋊C4), Q8⋊Q8⋊45C2, (C2×Q8).267D4, C2.61(Q8○D8), C4.47(C2×SD16), D4.33(C4○D4), C4⋊C8.349C22, C4⋊C4.427C23, (C2×C4).551C24, (C4×C8).278C22, (C2×C8).367C23, C4.SD16⋊31C2, (C4×SD16).12C2, C4⋊Q8.180C22, C2.59(Q8⋊5D4), D4.D4.12C2, (C4×D4).191C22, (C2×D4).428C23, (C4×Q8).305C22, (C2×Q8).250C23, C2.31(C22×SD16), C4.Q8.172C22, C22.811(C22×D4), D4⋊C4.217C22, Q8⋊C4.120C22, (C2×SD16).167C22, C4.252(C2×C4○D4), (C2×C4).1099(C2×D4), SmallGroup(128,2091)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊7SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd=c3 >
Subgroups: 328 in 185 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16, C22×Q8, C4×SD16, C8×Q8, D4.D4, Q8⋊Q8, C4.SD16, D4×Q8, Q8⋊3Q8, Q8⋊7SD16
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, Q8○D8, Q8⋊7SD16
►On 64 points
(1 30 51 47)(2 31 52 48)(3 32 53 41)(4 25 54 42)(5 26 55 43)(6 27 56 44)(7 28 49 45)(8 29 50 46)(9 62 37 21)(10 63 38 22)(11 64 39 23)(12 57 40 24)(13 58 33 17)(14 59 34 18)(15 60 35 19)(16 61 36 20)
(1 63 51 22)(2 23 52 64)(3 57 53 24)(4 17 54 58)(5 59 55 18)(6 19 56 60)(7 61 49 20)(8 21 50 62)(9 29 37 46)(10 47 38 30)(11 31 39 48)(12 41 40 32)(13 25 33 42)(14 43 34 26)(15 27 35 44)(16 45 36 28)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 45)(42 48)(44 46)(49 53)(50 56)(52 54)(57 61)(58 64)(60 62)
G:=sub<Sym(64)| (1,30,51,47)(2,31,52,48)(3,32,53,41)(4,25,54,42)(5,26,55,43)(6,27,56,44)(7,28,49,45)(8,29,50,46)(9,62,37,21)(10,63,38,22)(11,64,39,23)(12,57,40,24)(13,58,33,17)(14,59,34,18)(15,60,35,19)(16,61,36,20), (1,63,51,22)(2,23,52,64)(3,57,53,24)(4,17,54,58)(5,59,55,18)(6,19,56,60)(7,61,49,20)(8,21,50,62)(9,29,37,46)(10,47,38,30)(11,31,39,48)(12,41,40,32)(13,25,33,42)(14,43,34,26)(15,27,35,44)(16,45,36,28), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,45)(42,48)(44,46)(49,53)(50,56)(52,54)(57,61)(58,64)(60,62)>;
G:=Group( (1,30,51,47)(2,31,52,48)(3,32,53,41)(4,25,54,42)(5,26,55,43)(6,27,56,44)(7,28,49,45)(8,29,50,46)(9,62,37,21)(10,63,38,22)(11,64,39,23)(12,57,40,24)(13,58,33,17)(14,59,34,18)(15,60,35,19)(16,61,36,20), (1,63,51,22)(2,23,52,64)(3,57,53,24)(4,17,54,58)(5,59,55,18)(6,19,56,60)(7,61,49,20)(8,21,50,62)(9,29,37,46)(10,47,38,30)(11,31,39,48)(12,41,40,32)(13,25,33,42)(14,43,34,26)(15,27,35,44)(16,45,36,28), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,45)(42,48)(44,46)(49,53)(50,56)(52,54)(57,61)(58,64)(60,62) );
G=PermutationGroup([[(1,30,51,47),(2,31,52,48),(3,32,53,41),(4,25,54,42),(5,26,55,43),(6,27,56,44),(7,28,49,45),(8,29,50,46),(9,62,37,21),(10,63,38,22),(11,64,39,23),(12,57,40,24),(13,58,33,17),(14,59,34,18),(15,60,35,19),(16,61,36,20)], [(1,63,51,22),(2,23,52,64),(3,57,53,24),(4,17,54,58),(5,59,55,18),(6,19,56,60),(7,61,49,20),(8,21,50,62),(9,29,37,46),(10,47,38,30),(11,31,39,48),(12,41,40,32),(13,25,33,42),(14,43,34,26),(15,27,35,44),(16,45,36,28)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(25,31),(27,29),(28,32),(33,39),(35,37),(36,40),(41,45),(42,48),(44,46),(49,53),(50,56),(52,54),(57,61),(58,64),(60,62)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | ··· | 4M | 4N | ··· | 4S | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 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 | D4 | C4○D4 | SD16 | 2- 1+4 | Q8○D8 |
| kernel | Q8⋊7SD16 | C4×SD16 | C8×Q8 | D4.D4 | Q8⋊Q8 | C4.SD16 | D4×Q8 | Q8⋊3Q8 | C4⋊C4 | C2×Q8 | D4 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of Q8⋊7SD16 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 10 | 16 | 0 | 0 |
| 16 | 7 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 12 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[10,16,0,0,16,7,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,0,12,0,0,10,10],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,16] >;
Q8⋊7SD16 in GAP, Magma, Sage, TeX
Q_8\rtimes_7{\rm SD}_{16} % in TeX
G:=Group("Q8:7SD16"); // GroupNames label
G:=SmallGroup(128,2091);
// by ID
G=gap.SmallGroup(128,2091);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,346,80,4037,1027,124]);
// 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=a^2*b,b*d=d*b,d*c*d=c^3>;
// generators/relations