p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊1C8, C42.15Q8, C42.419D4, C42.622C23, C8⋊1(C2×C8), C8⋊2C8⋊7C2, C8⋊1C8⋊11C2, C4.18(C4⋊C8), C8⋊C4.4C4, C4.27(C22×C8), (C22×C4).11Q8, C4⋊C8.266C22, C23.48(C4⋊C4), C22.12(C4⋊C8), (C4×C8).137C22, C42.121(C2×C4), (C22×C4).667D4, (C4×M4(2)).1C2, (C2×C4).18M4(2), C4.42(C2×M4(2)), C4.137(C8⋊C22), (C2×M4(2)).10C4, C4.131(C8.C22), (C2×C42).224C22, C2.1(M4(2).C4), C2.1(M4(2)⋊C4), C42.12C4.29C2, C2.7(C2×C4⋊C8), (C2×C4⋊C8).18C2, (C2×C4).19(C2×C8), (C2×C8).56(C2×C4), C22.48(C2×C4⋊C4), (C2×C4).149(C2×Q8), (C2×C4).116(C4⋊C4), (C2×C4).1458(C2×D4), (C22×C4).246(C2×C4), (C2×C4).504(C22×C4), SmallGroup(128,297)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊1C8
G = < a,b,c | a8=b2=c8=1, bab=a5, cac-1=a-1, bc=cb >
Subgroups: 132 in 90 conjugacy classes, 62 normal (32 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C4×C8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C8⋊2C8, C8⋊1C8, C4×M4(2), C2×C4⋊C8, C42.12C4, M4(2)⋊1C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C8⋊C22, C8.C22, C2×C4⋊C8, M4(2)⋊C4, M4(2).C4, M4(2)⋊1C8
►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 43)(2 48)(3 45)(4 42)(5 47)(6 44)(7 41)(8 46)(9 59)(10 64)(11 61)(12 58)(13 63)(14 60)(15 57)(16 62)(17 25)(18 30)(19 27)(20 32)(21 29)(22 26)(23 31)(24 28)(33 51)(34 56)(35 53)(36 50)(37 55)(38 52)(39 49)(40 54)
(1 30 35 62 43 18 53 16)(2 29 36 61 44 17 54 15)(3 28 37 60 45 24 55 14)(4 27 38 59 46 23 56 13)(5 26 39 58 47 22 49 12)(6 25 40 57 48 21 50 11)(7 32 33 64 41 20 51 10)(8 31 34 63 42 19 52 9)
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), (1,43)(2,48)(3,45)(4,42)(5,47)(6,44)(7,41)(8,46)(9,59)(10,64)(11,61)(12,58)(13,63)(14,60)(15,57)(16,62)(17,25)(18,30)(19,27)(20,32)(21,29)(22,26)(23,31)(24,28)(33,51)(34,56)(35,53)(36,50)(37,55)(38,52)(39,49)(40,54), (1,30,35,62,43,18,53,16)(2,29,36,61,44,17,54,15)(3,28,37,60,45,24,55,14)(4,27,38,59,46,23,56,13)(5,26,39,58,47,22,49,12)(6,25,40,57,48,21,50,11)(7,32,33,64,41,20,51,10)(8,31,34,63,42,19,52,9)>;
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), (1,43)(2,48)(3,45)(4,42)(5,47)(6,44)(7,41)(8,46)(9,59)(10,64)(11,61)(12,58)(13,63)(14,60)(15,57)(16,62)(17,25)(18,30)(19,27)(20,32)(21,29)(22,26)(23,31)(24,28)(33,51)(34,56)(35,53)(36,50)(37,55)(38,52)(39,49)(40,54), (1,30,35,62,43,18,53,16)(2,29,36,61,44,17,54,15)(3,28,37,60,45,24,55,14)(4,27,38,59,46,23,56,13)(5,26,39,58,47,22,49,12)(6,25,40,57,48,21,50,11)(7,32,33,64,41,20,51,10)(8,31,34,63,42,19,52,9) );
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)], [(1,43),(2,48),(3,45),(4,42),(5,47),(6,44),(7,41),(8,46),(9,59),(10,64),(11,61),(12,58),(13,63),(14,60),(15,57),(16,62),(17,25),(18,30),(19,27),(20,32),(21,29),(22,26),(23,31),(24,28),(33,51),(34,56),(35,53),(36,50),(37,55),(38,52),(39,49),(40,54)], [(1,30,35,62,43,18,53,16),(2,29,36,61,44,17,54,15),(3,28,37,60,45,24,55,14),(4,27,38,59,46,23,56,13),(5,26,39,58,47,22,49,12),(6,25,40,57,48,21,50,11),(7,32,33,64,41,20,51,10),(8,31,34,63,42,19,52,9)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D4 | Q8 | M4(2) | C8⋊C22 | C8.C22 | M4(2).C4 |
| kernel | M4(2)⋊1C8 | C8⋊2C8 | C8⋊1C8 | C4×M4(2) | C2×C4⋊C8 | C42.12C4 | C8⋊C4 | C2×M4(2) | M4(2) | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 |
Matrix representation of M4(2)⋊1C8 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 1 | 16 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 | 16 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 16 | 8 | 0 | 0 |
| 0 | 0 | 7 | 10 | 7 | 16 |
| 0 | 0 | 0 | 10 | 16 | 10 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,16,0,0,0,15,16,1,1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,9,16,7,0,0,0,14,8,10,10,0,0,0,0,7,16,0,0,0,0,16,10] >;
M4(2)⋊1C8 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_1C_8
% in TeX
G:=Group("M4(2):1C8"); // GroupNames label
G:=SmallGroup(128,297);
// by ID
G=gap.SmallGroup(128,297);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,387,1123,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^8=1,b*a*b=a^5,c*a*c^-1=a^-1,b*c=c*b>;
// generators/relations