p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊5C8, C42.637C23, C8⋊3(C2×C8), D4⋊C8⋊5C2, (C8×D4)⋊2C2, D4⋊2(C2×C8), C8⋊C8⋊3C2, C2.9(C8×D4), C8⋊2C8⋊11C2, (C4×D8).15C2, (C2×D8).11C4, (C2×C8).306D4, C2.D8.19C4, C4.12(C8○D4), C4.12(C22×C8), (C4×C8).13C22, C22.82(C4×D4), D4⋊C4.10C4, C2.1(D8⋊C4), C2.3(C8.26D4), C4⋊C8.275C22, C4.142(C8⋊C22), (C4×D4).271C22, C4⋊C4.131(C2×C4), (C2×C8).125(C2×C4), (C2×D4).151(C2×C4), (C2×C4).1473(C2×D4), (C2×C4).498(C4○D4), (C2×C4).329(C22×C4), SmallGroup(128,312)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊5C8
G = < a,b,c | a8=b2=c8=1, bab=a-1, cac-1=a5, cbc-1=a4b >
Subgroups: 184 in 96 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, C22×C4, C2×D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C22×C8, C2×D8, C8⋊C8, D4⋊C8, C8⋊2C8, C8×D4, C4×D8, D8⋊5C8
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×D4, D8⋊C4, C8.26D4, D8⋊5C8
►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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 32)(9 35)(10 34)(11 33)(12 40)(13 39)(14 38)(15 37)(16 36)(17 49)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(41 60)(42 59)(43 58)(44 57)(45 64)(46 63)(47 62)(48 61)
(1 38 42 21 31 10 59 49)(2 35 43 18 32 15 60 54)(3 40 44 23 25 12 61 51)(4 37 45 20 26 9 62 56)(5 34 46 17 27 14 63 53)(6 39 47 22 28 11 64 50)(7 36 48 19 29 16 57 55)(8 33 41 24 30 13 58 52)
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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,32)(9,35)(10,34)(11,33)(12,40)(13,39)(14,38)(15,37)(16,36)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(41,60)(42,59)(43,58)(44,57)(45,64)(46,63)(47,62)(48,61), (1,38,42,21,31,10,59,49)(2,35,43,18,32,15,60,54)(3,40,44,23,25,12,61,51)(4,37,45,20,26,9,62,56)(5,34,46,17,27,14,63,53)(6,39,47,22,28,11,64,50)(7,36,48,19,29,16,57,55)(8,33,41,24,30,13,58,52)>;
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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,32)(9,35)(10,34)(11,33)(12,40)(13,39)(14,38)(15,37)(16,36)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(41,60)(42,59)(43,58)(44,57)(45,64)(46,63)(47,62)(48,61), (1,38,42,21,31,10,59,49)(2,35,43,18,32,15,60,54)(3,40,44,23,25,12,61,51)(4,37,45,20,26,9,62,56)(5,34,46,17,27,14,63,53)(6,39,47,22,28,11,64,50)(7,36,48,19,29,16,57,55)(8,33,41,24,30,13,58,52) );
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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,32),(9,35),(10,34),(11,33),(12,40),(13,39),(14,38),(15,37),(16,36),(17,49),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(41,60),(42,59),(43,58),(44,57),(45,64),(46,63),(47,62),(48,61)], [(1,38,42,21,31,10,59,49),(2,35,43,18,32,15,60,54),(3,40,44,23,25,12,61,51),(4,37,45,20,26,9,62,56),(5,34,46,17,27,14,63,53),(6,39,47,22,28,11,64,50),(7,36,48,19,29,16,57,55),(8,33,41,24,30,13,58,52)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 | C8⋊C22 | C8.26D4 |
| kernel | D8⋊5C8 | C8⋊C8 | D4⋊C8 | C8⋊2C8 | C8×D4 | C4×D8 | D4⋊C4 | C2.D8 | C2×D8 | D8 | C2×C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 16 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of D8⋊5C8 ►in GL6(𝔽17)
| 1 | 4 | 0 | 0 | 0 | 0 |
| 8 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 11 | 6 |
| 0 | 0 | 3 | 14 | 0 | 6 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 | 3 | 0 |
| 16 | 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 16 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 1 | 1 | 15 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [1,8,0,0,0,0,4,16,0,0,0,0,0,0,3,3,3,3,0,0,14,14,3,0,0,0,11,0,0,3,0,0,6,6,0,0],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,15,0,16] >;
D8⋊5C8 in GAP, Magma, Sage, TeX
D_8\rtimes_5C_8
% in TeX
G:=Group("D8:5C8"); // GroupNames label
G:=SmallGroup(128,312);
// by ID
G=gap.SmallGroup(128,312);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,268,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b>;
// generators/relations