p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊C8, C42.635C23, C8⋊2(C2×C8), Q8⋊C8⋊5C2, Q8⋊2(C2×C8), (C8×Q8)⋊2C2, C8⋊C8⋊2C2, C2.7(C8×D4), D4⋊C8.1C2, C8⋊1C8⋊15C2, D4.2(C2×C8), (C8×D4).3C2, C4.Q8.5C4, (C2×C8).304D4, D4⋊C4.9C4, C4.10(C8○D4), (C4×C8).11C22, C4.10(C22×C8), Q8⋊C4.9C4, (C4×SD16).3C2, (C2×SD16).1C4, C22.80(C4×D4), C2.1(C8.26D4), C4⋊C8.273C22, C4.141(C8⋊C22), (C4×D4).270C22, (C4×Q8).257C22, C2.1(SD16⋊C4), C4.135(C8.C22), C4⋊C4.129(C2×C4), (C2×C8).123(C2×C4), (C2×D4).150(C2×C4), (C2×C4).1471(C2×D4), (C2×Q8).133(C2×C4), (C2×C4).496(C4○D4), (C2×C4).327(C22×C4), SmallGroup(128,310)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊C8
G = < a,b,c | a8=b2=c8=1, bab=a3, cac-1=a5, bc=cb >
Subgroups: 152 in 88 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C4×C8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C8⋊1C8, C8×D4, C4×SD16, C8×Q8, SD16⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8⋊C22, C8.C22, C8×D4, SD16⋊C4, C8.26D4, SD16⋊C8
►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 64)(2 59)(3 62)(4 57)(5 60)(6 63)(7 58)(8 61)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)(17 46)(18 41)(19 44)(20 47)(21 42)(22 45)(23 48)(24 43)(33 50)(34 53)(35 56)(36 51)(37 54)(38 49)(39 52)(40 55)
(1 27 52 45 64 14 39 22)(2 32 53 42 57 11 40 19)(3 29 54 47 58 16 33 24)(4 26 55 44 59 13 34 21)(5 31 56 41 60 10 35 18)(6 28 49 46 61 15 36 23)(7 25 50 43 62 12 37 20)(8 30 51 48 63 9 38 17)
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), (1,64)(2,59)(3,62)(4,57)(5,60)(6,63)(7,58)(8,61)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)(17,46)(18,41)(19,44)(20,47)(21,42)(22,45)(23,48)(24,43)(33,50)(34,53)(35,56)(36,51)(37,54)(38,49)(39,52)(40,55), (1,27,52,45,64,14,39,22)(2,32,53,42,57,11,40,19)(3,29,54,47,58,16,33,24)(4,26,55,44,59,13,34,21)(5,31,56,41,60,10,35,18)(6,28,49,46,61,15,36,23)(7,25,50,43,62,12,37,20)(8,30,51,48,63,9,38,17)>;
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), (1,64)(2,59)(3,62)(4,57)(5,60)(6,63)(7,58)(8,61)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)(17,46)(18,41)(19,44)(20,47)(21,42)(22,45)(23,48)(24,43)(33,50)(34,53)(35,56)(36,51)(37,54)(38,49)(39,52)(40,55), (1,27,52,45,64,14,39,22)(2,32,53,42,57,11,40,19)(3,29,54,47,58,16,33,24)(4,26,55,44,59,13,34,21)(5,31,56,41,60,10,35,18)(6,28,49,46,61,15,36,23)(7,25,50,43,62,12,37,20)(8,30,51,48,63,9,38,17) );
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)], [(1,64),(2,59),(3,62),(4,57),(5,60),(6,63),(7,58),(8,61),(9,28),(10,31),(11,26),(12,29),(13,32),(14,27),(15,30),(16,25),(17,46),(18,41),(19,44),(20,47),(21,42),(22,45),(23,48),(24,43),(33,50),(34,53),(35,56),(36,51),(37,54),(38,49),(39,52),(40,55)], [(1,27,52,45,64,14,39,22),(2,32,53,42,57,11,40,19),(3,29,54,47,58,16,33,24),(4,26,55,44,59,13,34,21),(5,31,56,41,60,10,35,18),(6,28,49,46,61,15,36,23),(7,25,50,43,62,12,37,20),(8,30,51,48,63,9,38,17)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H | 8I | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 | C8⋊C22 | C8.C22 | C8.26D4 |
| kernel | SD16⋊C8 | C8⋊C8 | D4⋊C8 | Q8⋊C8 | C8⋊1C8 | C8×D4 | C4×SD16 | C8×Q8 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C2×SD16 | SD16 | C2×C8 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of SD16⋊C8 ►in GL6(𝔽17)
| 1 | 4 | 0 | 0 | 0 | 0 |
| 8 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 16 |
| 0 | 0 | 15 | 13 | 9 | 1 |
| 0 | 0 | 0 | 1 | 0 | 13 |
| 0 | 0 | 8 | 16 | 2 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,8,0,0,0,0,4,16,0,0,0,0,0,0,0,15,0,8,0,0,4,13,1,16,0,0,0,9,0,2,0,0,16,1,13,4],[16,9,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
SD16⋊C8 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes C_8 % in TeX
G:=Group("SD16:C8"); // GroupNames label
G:=SmallGroup(128,310);
// by ID
G=gap.SmallGroup(128,310);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^8=1,b*a*b=a^3,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations