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, (C4×C8).278C22, (C2×C4).551C24, (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
Subgroups: 328 in 185 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×4], C8 [×4], C2×C4, C2×C4 [×6], C2×C4 [×13], D4 [×2], D4, Q8 [×4], Q8 [×12], C23, C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×12], C2×C8, C2×C8 [×3], SD16 [×6], C22×C4 [×3], C2×D4, C2×Q8, C2×Q8 [×3], C2×Q8 [×7], C4×C8 [×3], D4⋊C4, Q8⋊C4 [×9], C4⋊C8 [×3], C4.Q8 [×3], C4×D4 [×3], C4×Q8, C4×Q8 [×3], C4×Q8, C22⋊Q8 [×3], C42.C2 [×3], C4⋊Q8 [×6], C2×SD16 [×3], C22×Q8, C4×SD16 [×3], C8×Q8, D4.D4 [×3], Q8⋊Q8 [×3], C4.SD16 [×3], D4×Q8, Q8⋊3Q8, Q8⋊7SD16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×SD16 [×6], C22×D4, C2×C4○D4, 2- (1+4), Q8⋊5D4, C22×SD16, Q8○D8, Q8⋊7SD16
Generators and relations
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 >
►On 64 points
(1 30 51 43)(2 31 52 44)(3 32 53 45)(4 25 54 46)(5 26 55 47)(6 27 56 48)(7 28 49 41)(8 29 50 42)(9 58 33 21)(10 59 34 22)(11 60 35 23)(12 61 36 24)(13 62 37 17)(14 63 38 18)(15 64 39 19)(16 57 40 20)
(1 63 51 18)(2 19 52 64)(3 57 53 20)(4 21 54 58)(5 59 55 22)(6 23 56 60)(7 61 49 24)(8 17 50 62)(9 25 33 46)(10 47 34 26)(11 27 35 48)(12 41 36 28)(13 29 37 42)(14 43 38 30)(15 31 39 44)(16 45 40 32)
(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)| (1,30,51,43)(2,31,52,44)(3,32,53,45)(4,25,54,46)(5,26,55,47)(6,27,56,48)(7,28,49,41)(8,29,50,42)(9,58,33,21)(10,59,34,22)(11,60,35,23)(12,61,36,24)(13,62,37,17)(14,63,38,18)(15,64,39,19)(16,57,40,20), (1,63,51,18)(2,19,52,64)(3,57,53,20)(4,21,54,58)(5,59,55,22)(6,23,56,60)(7,61,49,24)(8,17,50,62)(9,25,33,46)(10,47,34,26)(11,27,35,48)(12,41,36,28)(13,29,37,42)(14,43,38,30)(15,31,39,44)(16,45,40,32), (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( (1,30,51,43)(2,31,52,44)(3,32,53,45)(4,25,54,46)(5,26,55,47)(6,27,56,48)(7,28,49,41)(8,29,50,42)(9,58,33,21)(10,59,34,22)(11,60,35,23)(12,61,36,24)(13,62,37,17)(14,63,38,18)(15,64,39,19)(16,57,40,20), (1,63,51,18)(2,19,52,64)(3,57,53,20)(4,21,54,58)(5,59,55,22)(6,23,56,60)(7,61,49,24)(8,17,50,62)(9,25,33,46)(10,47,34,26)(11,27,35,48)(12,41,36,28)(13,29,37,42)(14,43,38,30)(15,31,39,44)(16,45,40,32), (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([(1,30,51,43),(2,31,52,44),(3,32,53,45),(4,25,54,46),(5,26,55,47),(6,27,56,48),(7,28,49,41),(8,29,50,42),(9,58,33,21),(10,59,34,22),(11,60,35,23),(12,61,36,24),(13,62,37,17),(14,63,38,18),(15,64,39,19),(16,57,40,20)], [(1,63,51,18),(2,19,52,64),(3,57,53,20),(4,21,54,58),(5,59,55,22),(6,23,56,60),(7,61,49,24),(8,17,50,62),(9,25,33,46),(10,47,34,26),(11,27,35,48),(12,41,36,28),(13,29,37,42),(14,43,38,30),(15,31,39,44),(16,45,40,32)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_7SD_{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