p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊6D8, C82⋊9C2, C8⋊4M4(2), C42.646C23, C2.5(C4×D8), D4⋊C8⋊35C2, (C2×D8).8C4, C8⋊1C8⋊30C2, (C4×D8).3C2, C4.87(C2×D8), C8⋊6D4⋊30C2, (C2×C8).284D4, C2.D8.13C4, D4⋊C4.1C4, C2.10(C8○D8), C4.13(C8○D4), C2.6(C8⋊6D4), C4.7(C2×M4(2)), C4.132(C4○D8), C4⋊C8.224C22, (C4×C8).392C22, (C4×D4).13C22, C22.137(C4×D4), C4⋊C4.59(C2×C4), (C2×C8).169(C2×C4), (C2×D4).58(C2×C4), (C2×C4).1482(C2×D4), (C2×C4).507(C4○D4), (C2×C4).338(C22×C4), SmallGroup(128,321)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊6D8
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a5, cbc=b-1 >
Subgroups: 184 in 90 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C2×M4(2), C2×D8, C82, D4⋊C8, C8⋊1C8, C8⋊6D4, C4×D8, C8⋊6D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×D8, C4○D8, C8⋊6D4, C4×D8, C8○D8, C8⋊6D8
►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 24 15 62 31 51 34 43)(2 17 16 63 32 52 35 44)(3 18 9 64 25 53 36 45)(4 19 10 57 26 54 37 46)(5 20 11 58 27 55 38 47)(6 21 12 59 28 56 39 48)(7 22 13 60 29 49 40 41)(8 23 14 61 30 50 33 42)
(1 43)(2 48)(3 45)(4 42)(5 47)(6 44)(7 41)(8 46)(9 53)(10 50)(11 55)(12 52)(13 49)(14 54)(15 51)(16 56)(17 39)(18 36)(19 33)(20 38)(21 35)(22 40)(23 37)(24 34)(25 64)(26 61)(27 58)(28 63)(29 60)(30 57)(31 62)(32 59)
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), (1,24,15,62,31,51,34,43)(2,17,16,63,32,52,35,44)(3,18,9,64,25,53,36,45)(4,19,10,57,26,54,37,46)(5,20,11,58,27,55,38,47)(6,21,12,59,28,56,39,48)(7,22,13,60,29,49,40,41)(8,23,14,61,30,50,33,42), (1,43)(2,48)(3,45)(4,42)(5,47)(6,44)(7,41)(8,46)(9,53)(10,50)(11,55)(12,52)(13,49)(14,54)(15,51)(16,56)(17,39)(18,36)(19,33)(20,38)(21,35)(22,40)(23,37)(24,34)(25,64)(26,61)(27,58)(28,63)(29,60)(30,57)(31,62)(32,59)>;
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), (1,24,15,62,31,51,34,43)(2,17,16,63,32,52,35,44)(3,18,9,64,25,53,36,45)(4,19,10,57,26,54,37,46)(5,20,11,58,27,55,38,47)(6,21,12,59,28,56,39,48)(7,22,13,60,29,49,40,41)(8,23,14,61,30,50,33,42), (1,43)(2,48)(3,45)(4,42)(5,47)(6,44)(7,41)(8,46)(9,53)(10,50)(11,55)(12,52)(13,49)(14,54)(15,51)(16,56)(17,39)(18,36)(19,33)(20,38)(21,35)(22,40)(23,37)(24,34)(25,64)(26,61)(27,58)(28,63)(29,60)(30,57)(31,62)(32,59) );
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)], [(1,24,15,62,31,51,34,43),(2,17,16,63,32,52,35,44),(3,18,9,64,25,53,36,45),(4,19,10,57,26,54,37,46),(5,20,11,58,27,55,38,47),(6,21,12,59,28,56,39,48),(7,22,13,60,29,49,40,41),(8,23,14,61,30,50,33,42)], [(1,43),(2,48),(3,45),(4,42),(5,47),(6,44),(7,41),(8,46),(9,53),(10,50),(11,55),(12,52),(13,49),(14,54),(15,51),(16,56),(17,39),(18,36),(19,33),(20,38),(21,35),(22,40),(23,37),(24,34),(25,64),(26,61),(27,58),(28,63),(29,60),(30,57),(31,62),(32,59)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8X | 8Y | 8Z | 8AA | 8AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | M4(2) | D8 | C4○D4 | C8○D4 | C4○D8 | C8○D8 |
| kernel | C8⋊6D8 | C82 | D4⋊C8 | C8⋊1C8 | C8⋊6D4 | C4×D8 | D4⋊C4 | C2.D8 | C2×D8 | C2×C8 | C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C8⋊6D8 ►in GL4(𝔽17) generated by
| 0 | 8 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 4 | 9 |
| 3 | 3 | 0 | 0 |
| 14 | 3 | 0 | 0 |
| 0 | 0 | 16 | 4 |
| 0 | 0 | 8 | 1 |
| 3 | 3 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 0 | 0 | 16 | 4 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [0,9,0,0,8,0,0,0,0,0,8,4,0,0,2,9],[3,14,0,0,3,3,0,0,0,0,16,8,0,0,4,1],[3,3,0,0,3,14,0,0,0,0,16,0,0,0,4,1] >;
C8⋊6D8 in GAP, Magma, Sage, TeX
C_8\rtimes_6D_8
% in TeX
G:=Group("C8:6D8"); // GroupNames label
G:=SmallGroup(128,321);
// by ID
G=gap.SmallGroup(128,321);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,723,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations