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 [×3], C2, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×3], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×5], C23, C42 [×4], C2×C8 [×4], C2×C8 [×4], M4(2) [×6], C22×C4 [×3], C4×C8 [×2], C8⋊C4 [×2], C8⋊C4, C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C2×M4(2) [×2], C2×M4(2), C8⋊2C8 [×2], C8⋊1C8 [×2], C4×M4(2), C4⋊M4(2), C42.6C4, C8⋊1M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C2×M4(2) [×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 16 48 26 39 60 54 19)(2 11 41 29 40 63 55 22)(3 14 42 32 33 58 56 17)(4 9 43 27 34 61 49 20)(5 12 44 30 35 64 50 23)(6 15 45 25 36 59 51 18)(7 10 46 28 37 62 52 21)(8 13 47 31 38 57 53 24)
(2 6)(4 8)(9 57)(10 62)(11 59)(12 64)(13 61)(14 58)(15 63)(16 60)(17 32)(18 29)(19 26)(20 31)(21 28)(22 25)(23 30)(24 27)(34 38)(36 40)(41 45)(43 47)(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), (1,16,48,26,39,60,54,19)(2,11,41,29,40,63,55,22)(3,14,42,32,33,58,56,17)(4,9,43,27,34,61,49,20)(5,12,44,30,35,64,50,23)(6,15,45,25,36,59,51,18)(7,10,46,28,37,62,52,21)(8,13,47,31,38,57,53,24), (2,6)(4,8)(9,57)(10,62)(11,59)(12,64)(13,61)(14,58)(15,63)(16,60)(17,32)(18,29)(19,26)(20,31)(21,28)(22,25)(23,30)(24,27)(34,38)(36,40)(41,45)(43,47)(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), (1,16,48,26,39,60,54,19)(2,11,41,29,40,63,55,22)(3,14,42,32,33,58,56,17)(4,9,43,27,34,61,49,20)(5,12,44,30,35,64,50,23)(6,15,45,25,36,59,51,18)(7,10,46,28,37,62,52,21)(8,13,47,31,38,57,53,24), (2,6)(4,8)(9,57)(10,62)(11,59)(12,64)(13,61)(14,58)(15,63)(16,60)(17,32)(18,29)(19,26)(20,31)(21,28)(22,25)(23,30)(24,27)(34,38)(36,40)(41,45)(43,47)(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)], [(1,16,48,26,39,60,54,19),(2,11,41,29,40,63,55,22),(3,14,42,32,33,58,56,17),(4,9,43,27,34,61,49,20),(5,12,44,30,35,64,50,23),(6,15,45,25,36,59,51,18),(7,10,46,28,37,62,52,21),(8,13,47,31,38,57,53,24)], [(2,6),(4,8),(9,57),(10,62),(11,59),(12,64),(13,61),(14,58),(15,63),(16,60),(17,32),(18,29),(19,26),(20,31),(21,28),(22,25),(23,30),(24,27),(34,38),(36,40),(41,45),(43,47),(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