p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊4Q8, C42.522C23, C4.962- 1+4, C4⋊C4○3SD16, (C8×Q8)⋊15C2, Q8.8(C2×Q8), C2.39(D4×Q8), C8.38(C2×Q8), C4⋊C4.416D4, Q8⋊3Q8⋊9C2, C8⋊3Q8⋊28C2, Q8.Q8⋊11C2, D4.10(C2×Q8), D4.Q8.1C2, C4.Q16⋊17C2, C4.77(C4○D8), (C2×Q8).187D4, D4⋊3Q8.9C2, C8.5Q8⋊11C2, C4.39(C22×Q8), C4⋊C4.270C23, C4⋊C8.328C22, (C2×C4).573C24, (C4×C8).124C22, (C2×C8).374C23, D4⋊Q8.11C2, (C4×SD16).15C2, C4⋊Q8.202C22, C2.D8.72C22, (C4×D4).211C22, (C2×D4).436C23, (C2×Q8).408C23, (C4×Q8).312C22, C4.Q8.116C22, C2.107(D4○SD16), C22.833(C22×D4), C42.C2.72C22, D4⋊C4.173C22, Q8⋊C4.186C22, (C2×SD16).180C22, C2.77(C2×C4○D8), (C2×C4).179(C2×D4), SmallGroup(128,2113)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊4Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, ad=da, bc=cb, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 288 in 169 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C2×SD16, C4×SD16, C4×SD16, C8×Q8, D4⋊Q8, C4.Q16, D4.Q8, Q8.Q8, C8⋊3Q8, C8.5Q8, D4⋊3Q8, Q8⋊3Q8, SD16⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4○D8, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C4○D8, D4○SD16, SD16⋊4Q8
►On 64 points
(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)
(1 9 22 51)(2 10 23 52)(3 11 24 53)(4 12 17 54)(5 13 18 55)(6 14 19 56)(7 15 20 49)(8 16 21 50)(25 37 58 45)(26 38 59 46)(27 39 60 47)(28 40 61 48)(29 33 62 41)(30 34 63 42)(31 35 64 43)(32 36 57 44)
(1 36 22 44)(2 37 23 45)(3 38 24 46)(4 39 17 47)(5 40 18 48)(6 33 19 41)(7 34 20 42)(8 35 21 43)(9 32 51 57)(10 25 52 58)(11 26 53 59)(12 27 54 60)(13 28 55 61)(14 29 56 62)(15 30 49 63)(16 31 50 64)
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), (1,18)(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), (1,9,22,51)(2,10,23,52)(3,11,24,53)(4,12,17,54)(5,13,18,55)(6,14,19,56)(7,15,20,49)(8,16,21,50)(25,37,58,45)(26,38,59,46)(27,39,60,47)(28,40,61,48)(29,33,62,41)(30,34,63,42)(31,35,64,43)(32,36,57,44), (1,36,22,44)(2,37,23,45)(3,38,24,46)(4,39,17,47)(5,40,18,48)(6,33,19,41)(7,34,20,42)(8,35,21,43)(9,32,51,57)(10,25,52,58)(11,26,53,59)(12,27,54,60)(13,28,55,61)(14,29,56,62)(15,30,49,63)(16,31,50,64)>;
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), (1,18)(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), (1,9,22,51)(2,10,23,52)(3,11,24,53)(4,12,17,54)(5,13,18,55)(6,14,19,56)(7,15,20,49)(8,16,21,50)(25,37,58,45)(26,38,59,46)(27,39,60,47)(28,40,61,48)(29,33,62,41)(30,34,63,42)(31,35,64,43)(32,36,57,44), (1,36,22,44)(2,37,23,45)(3,38,24,46)(4,39,17,47)(5,40,18,48)(6,33,19,41)(7,34,20,42)(8,35,21,43)(9,32,51,57)(10,25,52,58)(11,26,53,59)(12,27,54,60)(13,28,55,61)(14,29,56,62)(15,30,49,63)(16,31,50,64) );
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)], [(1,18),(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)], [(1,9,22,51),(2,10,23,52),(3,11,24,53),(4,12,17,54),(5,13,18,55),(6,14,19,56),(7,15,20,49),(8,16,21,50),(25,37,58,45),(26,38,59,46),(27,39,60,47),(28,40,61,48),(29,33,62,41),(30,34,63,42),(31,35,64,43),(32,36,57,44)], [(1,36,22,44),(2,37,23,45),(3,38,24,46),(4,39,17,47),(5,40,18,48),(6,33,19,41),(7,34,20,42),(8,35,21,43),(9,32,51,57),(10,25,52,58),(11,26,53,59),(12,27,54,60),(13,28,55,61),(14,29,56,62),(15,30,49,63),(16,31,50,64)]])
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 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C4○D8 | 2- 1+4 | D4○SD16 |
| kernel | SD16⋊4Q8 | C4×SD16 | C8×Q8 | D4⋊Q8 | C4.Q16 | D4.Q8 | Q8.Q8 | C8⋊3Q8 | C8.5Q8 | D4⋊3Q8 | Q8⋊3Q8 | C4⋊C4 | SD16 | C2×Q8 | C4 | C4 | C2 |
| # reps | 1 | 3 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 3 | 4 | 1 | 8 | 1 | 2 |
Matrix representation of SD16⋊4Q8 ►in GL4(𝔽17) generated by
| 7 | 7 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 16 | 16 |
| 4 | 8 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 13 | 13 |
G:=sub<GL(4,GF(17))| [7,5,0,0,7,0,0,0,0,0,16,0,0,0,0,16],[16,1,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,1,16,0,0,2,16],[4,13,0,0,8,13,0,0,0,0,4,13,0,0,0,13] >;
SD16⋊4Q8 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_4Q_8 % in TeX
G:=Group("SD16:4Q8"); // GroupNames label
G:=SmallGroup(128,2113);
// by ID
G=gap.SmallGroup(128,2113);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,568,758,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations