direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8×SD16, C42.520C23, C4.942- (1+4), C8⋊9(C2×Q8), (Q82)⋊6C2, Q8⋊4(C2×Q8), (C8×Q8)⋊23C2, D4.8(C2×Q8), (D4×Q8).9C2, C2.37(D4×Q8), C4⋊C4.281D4, C8⋊3Q8⋊27C2, Q8⋊Q8⋊47C2, (C2×Q8).271D4, C4.49(C2×SD16), C4.37(C22×Q8), C4⋊C4.268C23, C4⋊C8.351C22, (C2×C8).373C23, (C2×C4).571C24, (C4×C8).282C22, D4⋊2Q8.13C2, (C4×SD16).14C2, C4⋊Q8.200C22, (C2×D4).434C23, (C4×D4).209C22, (C4×Q8).201C22, (C2×Q8).407C23, C2.33(C22×SD16), C4.Q8.114C22, C2.106(D4○SD16), C22.831(C22×D4), D4⋊C4.189C22, Q8⋊C4.185C22, (C2×SD16).179C22, (C2×C4).1103(C2×D4), SmallGroup(128,2111)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 336 in 185 conjugacy classes, 104 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×11], C22, C22 [×4], C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×13], D4 [×2], D4, Q8 [×6], Q8 [×10], C23, C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×9], C4⋊C4 [×9], C2×C8, C2×C8 [×3], SD16 [×4], C22×C4 [×3], C2×D4, C2×Q8 [×2], C2×Q8 [×10], C4×C8 [×3], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8 [×3], C4.Q8 [×9], C4×D4 [×3], C4×Q8, C4×Q8 [×3], C4×Q8, C22⋊Q8 [×3], C4⋊Q8 [×6], C4⋊Q8 [×3], C2×SD16, C22×Q8, C4×SD16 [×3], C8×Q8, Q8⋊Q8 [×3], D4⋊2Q8 [×3], C8⋊3Q8 [×3], D4×Q8, Q82, Q8×SD16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C2×Q8 [×6], C24, C2×SD16 [×6], C22×D4, C22×Q8, 2- (1+4), D4×Q8, C22×SD16, D4○SD16, Q8×SD16
Generators and relations
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
►On 64 points
(1 13 22 55)(2 14 23 56)(3 15 24 49)(4 16 17 50)(5 9 18 51)(6 10 19 52)(7 11 20 53)(8 12 21 54)(25 33 58 41)(26 34 59 42)(27 35 60 43)(28 36 61 44)(29 37 62 45)(30 38 63 46)(31 39 64 47)(32 40 57 48)
(1 34 22 42)(2 35 23 43)(3 36 24 44)(4 37 17 45)(5 38 18 46)(6 39 19 47)(7 40 20 48)(8 33 21 41)(9 30 51 63)(10 31 52 64)(11 32 53 57)(12 25 54 58)(13 26 55 59)(14 27 56 60)(15 28 49 61)(16 29 50 62)
(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 18)(2 21)(3 24)(4 19)(5 22)(6 17)(7 20)(8 23)(9 55)(10 50)(11 53)(12 56)(13 51)(14 54)(15 49)(16 52)(25 60)(26 63)(27 58)(28 61)(29 64)(30 59)(31 62)(32 57)(33 43)(34 46)(35 41)(36 44)(37 47)(38 42)(39 45)(40 48)
G:=sub<Sym(64)| (1,13,22,55)(2,14,23,56)(3,15,24,49)(4,16,17,50)(5,9,18,51)(6,10,19,52)(7,11,20,53)(8,12,21,54)(25,33,58,41)(26,34,59,42)(27,35,60,43)(28,36,61,44)(29,37,62,45)(30,38,63,46)(31,39,64,47)(32,40,57,48), (1,34,22,42)(2,35,23,43)(3,36,24,44)(4,37,17,45)(5,38,18,46)(6,39,19,47)(7,40,20,48)(8,33,21,41)(9,30,51,63)(10,31,52,64)(11,32,53,57)(12,25,54,58)(13,26,55,59)(14,27,56,60)(15,28,49,61)(16,29,50,62), (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), (1,18)(2,21)(3,24)(4,19)(5,22)(6,17)(7,20)(8,23)(9,55)(10,50)(11,53)(12,56)(13,51)(14,54)(15,49)(16,52)(25,60)(26,63)(27,58)(28,61)(29,64)(30,59)(31,62)(32,57)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48)>;
G:=Group( (1,13,22,55)(2,14,23,56)(3,15,24,49)(4,16,17,50)(5,9,18,51)(6,10,19,52)(7,11,20,53)(8,12,21,54)(25,33,58,41)(26,34,59,42)(27,35,60,43)(28,36,61,44)(29,37,62,45)(30,38,63,46)(31,39,64,47)(32,40,57,48), (1,34,22,42)(2,35,23,43)(3,36,24,44)(4,37,17,45)(5,38,18,46)(6,39,19,47)(7,40,20,48)(8,33,21,41)(9,30,51,63)(10,31,52,64)(11,32,53,57)(12,25,54,58)(13,26,55,59)(14,27,56,60)(15,28,49,61)(16,29,50,62), (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), (1,18)(2,21)(3,24)(4,19)(5,22)(6,17)(7,20)(8,23)(9,55)(10,50)(11,53)(12,56)(13,51)(14,54)(15,49)(16,52)(25,60)(26,63)(27,58)(28,61)(29,64)(30,59)(31,62)(32,57)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48) );
G=PermutationGroup([(1,13,22,55),(2,14,23,56),(3,15,24,49),(4,16,17,50),(5,9,18,51),(6,10,19,52),(7,11,20,53),(8,12,21,54),(25,33,58,41),(26,34,59,42),(27,35,60,43),(28,36,61,44),(29,37,62,45),(30,38,63,46),(31,39,64,47),(32,40,57,48)], [(1,34,22,42),(2,35,23,43),(3,36,24,44),(4,37,17,45),(5,38,18,46),(6,39,19,47),(7,40,20,48),(8,33,21,41),(9,30,51,63),(10,31,52,64),(11,32,53,57),(12,25,54,58),(13,26,55,59),(14,27,56,60),(15,28,49,61),(16,29,50,62)], [(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)], [(1,18),(2,21),(3,24),(4,19),(5,22),(6,17),(7,20),(8,23),(9,55),(10,50),(11,53),(12,56),(13,51),(14,54),(15,49),(16,52),(25,60),(26,63),(27,58),(28,61),(29,64),(30,59),(31,62),(32,57),(33,43),(34,46),(35,41),(36,44),(37,47),(38,42),(39,45),(40,48)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 15 |
| 0 | 0 | 5 | 14 |
| 5 | 12 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,2,13],[1,0,0,0,0,1,0,0,0,0,3,5,0,0,15,14],[5,5,0,0,12,5,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16] >;
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 | Q8 | D4 | SD16 | 2- (1+4) | D4○SD16 |
| kernel | Q8×SD16 | C4×SD16 | C8×Q8 | Q8⋊Q8 | D4⋊2Q8 | C8⋊3Q8 | D4×Q8 | Q82 | C4⋊C4 | SD16 | C2×Q8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 4 | 1 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_8\times SD_{16} % in TeX
G:=Group("Q8xSD16"); // GroupNames label
G:=SmallGroup(128,2111);
// by ID
G=gap.SmallGroup(128,2111);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,346,80,4037,1027,124]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations