p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊8M4(2), C42.50Q8, C42.319D4, C42.623C23, (C4×C8).18C4, C8⋊2C8⋊24C2, (C22×C8).40C4, (C2×C4).69SD16, C4.98(C2×SD16), C4.10(C4.Q8), C4.5(C8.C4), (C22×C4).81Q8, C4⋊C8.214C22, C23.49(C4⋊C4), (C4×C8).424C22, C42.310(C2×C4), (C22×C4).574D4, C4.43(C2×M4(2)), C22.10(C4.Q8), C4⋊M4(2).20C2, C2.6(C4⋊M4(2)), (C2×C42).1041C22, (C2×C4×C8).52C2, C2.4(C2×C4.Q8), (C2×C4).74(C4⋊C4), (C2×C8).231(C2×C4), C2.6(C2×C8.C4), C22.80(C2×C4⋊C4), (C2×C4).150(C2×Q8), (C2×C4).1459(C2×D4), (C2×C4).505(C22×C4), (C22×C4).474(C2×C4), SmallGroup(128,298)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊8M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a3, ac=ca, cbc=b5 >
Subgroups: 140 in 92 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C4×C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C8⋊2C8, C2×C4×C8, C4⋊M4(2), C8⋊8M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), SD16, C22×C4, C2×D4, C2×Q8, C4.Q8, C8.C4, C2×C4⋊C4, C2×M4(2), C2×SD16, C4⋊M4(2), C2×C4.Q8, C2×C8.C4, C8⋊8M4(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 24 47 28 35 60 54 9)(2 19 48 31 36 63 55 12)(3 22 41 26 37 58 56 15)(4 17 42 29 38 61 49 10)(5 20 43 32 39 64 50 13)(6 23 44 27 40 59 51 16)(7 18 45 30 33 62 52 11)(8 21 46 25 34 57 53 14)
(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)
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), (1,24,47,28,35,60,54,9)(2,19,48,31,36,63,55,12)(3,22,41,26,37,58,56,15)(4,17,42,29,38,61,49,10)(5,20,43,32,39,64,50,13)(6,23,44,27,40,59,51,16)(7,18,45,30,33,62,52,11)(8,21,46,25,34,57,53,14), (9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)>;
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), (1,24,47,28,35,60,54,9)(2,19,48,31,36,63,55,12)(3,22,41,26,37,58,56,15)(4,17,42,29,38,61,49,10)(5,20,43,32,39,64,50,13)(6,23,44,27,40,59,51,16)(7,18,45,30,33,62,52,11)(8,21,46,25,34,57,53,14), (9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60) );
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)], [(1,24,47,28,35,60,54,9),(2,19,48,31,36,63,55,12),(3,22,41,26,37,58,56,15),(4,17,42,29,38,61,49,10),(5,20,43,32,39,64,50,13),(6,23,44,27,40,59,51,16),(7,18,45,30,33,62,52,11),(8,21,46,25,34,57,53,14)], [(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(17,61),(18,62),(19,63),(20,64),(21,57),(22,58),(23,59),(24,60)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | Q8 | M4(2) | SD16 | C8.C4 |
| kernel | C8⋊8M4(2) | C8⋊2C8 | C2×C4×C8 | C4⋊M4(2) | C4×C8 | C22×C8 | C42 | C42 | C22×C4 | C22×C4 | C8 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 8 | 8 | 8 |
Matrix representation of C8⋊8M4(2) ►in GL4(𝔽17) generated by
| 9 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [9,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,1,0,0,0,0,0,0,13,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
C8⋊8M4(2) in GAP, Magma, Sage, TeX
C_8\rtimes_8M_4(2)
% in TeX
G:=Group("C8:8M4(2)"); // GroupNames label
G:=SmallGroup(128,298);
// by ID
G=gap.SmallGroup(128,298);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1430,1123,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^3,a*c=c*a,c*b*c=b^5>;
// generators/relations