p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊1M4(2), C42.89D4, C42.16Q8, C42.626C23, C8⋊2C8⋊8C2, C8⋊1C8⋊12C2, C8⋊C4.14C4, (C22×C4).28Q8, C4⋊C8.217C22, C23.19(C4⋊C4), (C4×C8).138C22, C42.122(C2×C4), (C22×C4).249D4, (C4×M4(2)).2C2, C4.46(C2×M4(2)), C4.138(C8⋊C22), (C2×M4(2)).11C4, C4.132(C8.C22), C4⋊M4(2).22C2, C2.9(C4⋊M4(2)), C42.6C4.29C2, (C2×C42).225C22, C2.4(M4(2).C4), C2.4(M4(2)⋊C4), (C2×C4).33(C4⋊C4), (C2×C8).122(C2×C4), C22.83(C2×C4⋊C4), (C2×C4).153(C2×Q8), (C2×C4).1462(C2×D4), (C2×C4).508(C22×C4), (C22×C4).247(C2×C4), SmallGroup(128,301)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊1M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a5, cbc=b5 >
Subgroups: 132 in 83 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×M4(2), C2×M4(2), C8⋊2C8, C8⋊1C8, C4×M4(2), C4⋊M4(2), C42.6C4, C8⋊1M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C2×M4(2), C8⋊C22, C8.C22, C4⋊M4(2), M4(2)⋊C4, M4(2).C4, C8⋊1M4(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 22 43 26 39 60 52 13)(2 17 44 29 40 63 53 16)(3 20 45 32 33 58 54 11)(4 23 46 27 34 61 55 14)(5 18 47 30 35 64 56 9)(6 21 48 25 36 59 49 12)(7 24 41 28 37 62 50 15)(8 19 42 31 38 57 51 10)
(2 6)(4 8)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)(17 59)(18 64)(19 61)(20 58)(21 63)(22 60)(23 57)(24 62)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)
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), (1,22,43,26,39,60,52,13)(2,17,44,29,40,63,53,16)(3,20,45,32,33,58,54,11)(4,23,46,27,34,61,55,14)(5,18,47,30,35,64,56,9)(6,21,48,25,36,59,49,12)(7,24,41,28,37,62,50,15)(8,19,42,31,38,57,51,10), (2,6)(4,8)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25)(17,59)(18,64)(19,61)(20,58)(21,63)(22,60)(23,57)(24,62)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)>;
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), (1,22,43,26,39,60,52,13)(2,17,44,29,40,63,53,16)(3,20,45,32,33,58,54,11)(4,23,46,27,34,61,55,14)(5,18,47,30,35,64,56,9)(6,21,48,25,36,59,49,12)(7,24,41,28,37,62,50,15)(8,19,42,31,38,57,51,10), (2,6)(4,8)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25)(17,59)(18,64)(19,61)(20,58)(21,63)(22,60)(23,57)(24,62)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55) );
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)], [(1,22,43,26,39,60,52,13),(2,17,44,29,40,63,53,16),(3,20,45,32,33,58,54,11),(4,23,46,27,34,61,55,14),(5,18,47,30,35,64,56,9),(6,21,48,25,36,59,49,12),(7,24,41,28,37,62,50,15),(8,19,42,31,38,57,51,10)], [(2,6),(4,8),(9,30),(10,27),(11,32),(12,29),(13,26),(14,31),(15,28),(16,25),(17,59),(18,64),(19,61),(20,58),(21,63),(22,60),(23,57),(24,62),(34,38),(36,40),(42,46),(44,48),(49,53),(51,55)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | ··· | 8H | 8I | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 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 | D4 | Q8 | D4 | Q8 | M4(2) | C8⋊C22 | C8.C22 | M4(2).C4 |
| kernel | C8⋊1M4(2) | C8⋊2C8 | C8⋊1C8 | C4×M4(2) | C4⋊M4(2) | C42.6C4 | C8⋊C4 | C2×M4(2) | C42 | C42 | C22×C4 | C22×C4 | C8 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 |
Matrix representation of C8⋊1M4(2) ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 9 | 13 | 13 | 13 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 6 | 0 | 6 |
| 0 | 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 5 | 0 | 11 | 11 |
| 0 | 0 | 5 | 5 | 11 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,9,0,0,4,0,13,13,0,0,0,0,0,13,0,0,0,4,0,13],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,12,0,5,5,0,0,6,0,0,5,0,0,0,6,11,11,0,0,6,11,11,11],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,16,0,0,0,1,0,0,16] >;
C8⋊1M4(2) in GAP, Magma, Sage, TeX
C_8\rtimes_1M_4(2)
% in TeX
G:=Group("C8:1M4(2)"); // GroupNames label
G:=SmallGroup(128,301);
// by ID
G=gap.SmallGroup(128,301);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1430,387,1123,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^5,c*b*c=b^5>;
// generators/relations