p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊3M4(2), C42.651C23, C8⋊C8⋊8C2, D4⋊C8⋊36C2, C8⋊2C8⋊13C2, C8⋊6D4⋊31C2, (C4×D8).16C2, (C2×D8).12C4, (C2×C8).185D4, C2.D8.21C4, C4.18(C8○D4), D4⋊C4.12C4, C2.5(D8⋊C4), C2.11(C8⋊6D4), C4⋊C8.229C22, (C4×C8).320C22, (C4×D4).16C22, C22.142(C4×D4), C4.12(C2×M4(2)), C2.12(C8.26D4), C4.147(C8⋊C22), C4⋊C4.64(C2×C4), (C2×C8).59(C2×C4), (C2×D4).61(C2×C4), (C2×C4).1487(C2×D4), (C2×C4).512(C4○D4), (C2×C4).343(C22×C4), SmallGroup(128,326)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊3M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a-1, cbc=b5 >
Subgroups: 184 in 88 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, 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, C8⋊C8, D4⋊C8, C8⋊2C8, C8⋊6D4, C4×D8, C8⋊3M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22, C8⋊6D4, D8⋊C4, C8.26D4, C8⋊3M4(2)
►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 25 63 13 40 20 50 41)(2 28 64 16 33 23 51 44)(3 31 57 11 34 18 52 47)(4 26 58 14 35 21 53 42)(5 29 59 9 36 24 54 45)(6 32 60 12 37 19 55 48)(7 27 61 15 38 22 56 43)(8 30 62 10 39 17 49 46)
(2 8)(3 7)(4 6)(9 45)(10 44)(11 43)(12 42)(13 41)(14 48)(15 47)(16 46)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(33 39)(34 38)(35 37)(49 51)(52 56)(53 55)(57 61)(58 60)(62 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), (1,25,63,13,40,20,50,41)(2,28,64,16,33,23,51,44)(3,31,57,11,34,18,52,47)(4,26,58,14,35,21,53,42)(5,29,59,9,36,24,54,45)(6,32,60,12,37,19,55,48)(7,27,61,15,38,22,56,43)(8,30,62,10,39,17,49,46), (2,8)(3,7)(4,6)(9,45)(10,44)(11,43)(12,42)(13,41)(14,48)(15,47)(16,46)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,39)(34,38)(35,37)(49,51)(52,56)(53,55)(57,61)(58,60)(62,64)>;
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), (1,25,63,13,40,20,50,41)(2,28,64,16,33,23,51,44)(3,31,57,11,34,18,52,47)(4,26,58,14,35,21,53,42)(5,29,59,9,36,24,54,45)(6,32,60,12,37,19,55,48)(7,27,61,15,38,22,56,43)(8,30,62,10,39,17,49,46), (2,8)(3,7)(4,6)(9,45)(10,44)(11,43)(12,42)(13,41)(14,48)(15,47)(16,46)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,39)(34,38)(35,37)(49,51)(52,56)(53,55)(57,61)(58,60)(62,64) );
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)], [(1,25,63,13,40,20,50,41),(2,28,64,16,33,23,51,44),(3,31,57,11,34,18,52,47),(4,26,58,14,35,21,53,42),(5,29,59,9,36,24,54,45),(6,32,60,12,37,19,55,48),(7,27,61,15,38,22,56,43),(8,30,62,10,39,17,49,46)], [(2,8),(3,7),(4,6),(9,45),(10,44),(11,43),(12,42),(13,41),(14,48),(15,47),(16,46),(17,28),(18,27),(19,26),(20,25),(21,32),(22,31),(23,30),(24,29),(33,39),(34,38),(35,37),(49,51),(52,56),(53,55),(57,61),(58,60),(62,64)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8L | 8M | 8N | 8O | 8P |
| 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 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | M4(2) | C4○D4 | C8○D4 | C8⋊C22 | C8.26D4 |
| kernel | C8⋊3M4(2) | C8⋊C8 | D4⋊C8 | C8⋊2C8 | C8⋊6D4 | C4×D8 | D4⋊C4 | C2.D8 | C2×D8 | C2×C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 |
Matrix representation of C8⋊3M4(2) ►in GL6(𝔽17)
| 0 | 15 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 6 | 11 |
| 0 | 0 | 15 | 15 | 6 | 6 |
| 0 | 0 | 14 | 1 | 2 | 4 |
| 0 | 0 | 1 | 14 | 2 | 13 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 15 | 0 |
| 0 | 0 | 16 | 0 | 0 | 2 |
| 0 | 0 | 16 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [0,9,0,0,0,0,15,0,0,0,0,0,0,0,4,15,14,1,0,0,13,15,1,14,0,0,6,6,2,2,0,0,11,6,4,13],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,0,0,1,0,0,1,0,0,15,0,0,16,0,0,0,2,1,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
C8⋊3M4(2) in GAP, Magma, Sage, TeX
C_8\rtimes_3M_4(2)
% in TeX
G:=Group("C8:3M4(2)"); // GroupNames label
G:=SmallGroup(128,326);
// by ID
G=gap.SmallGroup(128,326);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,723,268,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^-1,c*b*c=b^5>;
// generators/relations