p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊7M4(2), C42.51Q8, C42.320D4, C42.624C23, (C4×C8).19C4, C8⋊1C8⋊25C2, (C2×C4).72D8, C4.83(C2×D8), (C2×C4).34Q16, C4.55(C2×Q16), (C22×C8).34C4, C4.14(C2.D8), C4.6(C8.C4), (C22×C4).82Q8, C4⋊C8.215C22, C23.50(C4⋊C4), C42.311(C2×C4), (C4×C8).390C22, (C22×C4).575D4, C4.44(C2×M4(2)), C22.12(C2.D8), C4⋊M4(2).21C2, C2.7(C4⋊M4(2)), (C2×C42).1042C22, (C2×C4×C8).32C2, C2.4(C2×C2.D8), (C2×C4).75(C4⋊C4), (C2×C8).220(C2×C4), C2.7(C2×C8.C4), C22.81(C2×C4⋊C4), (C2×C4).151(C2×Q8), (C2×C4).1460(C2×D4), (C2×C4).506(C22×C4), (C22×C4).475(C2×C4), SmallGroup(128,299)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊7M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >
Subgroups: 140 in 92 conjugacy classes, 60 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×4], C2×C8 [×8], M4(2) [×4], C22×C4 [×3], C4×C8 [×2], C4×C8 [×2], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C8⋊1C8 [×4], C2×C4×C8, C4⋊M4(2) [×2], C8⋊7M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, C2.D8 [×4], C8.C4 [×2], C2×C4⋊C4, C2×M4(2) [×2], C2×D8, C2×Q16, C4⋊M4(2), C2×C2.D8, C2×C8.C4, C8⋊7M4(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 21 44 25 40 59 53 12)(3 20 45 32 33 58 54 11)(4 19 46 31 34 57 55 10)(5 18 47 30 35 64 56 9)(6 17 48 29 36 63 49 16)(7 24 41 28 37 62 50 15)(8 23 42 27 38 61 51 14)
(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)
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)(2,21,44,25,40,59,53,12)(3,20,45,32,33,58,54,11)(4,19,46,31,34,57,55,10)(5,18,47,30,35,64,56,9)(6,17,48,29,36,63,49,16)(7,24,41,28,37,62,50,15)(8,23,42,27,38,61,51,14), (9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)>;
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)(2,21,44,25,40,59,53,12)(3,20,45,32,33,58,54,11)(4,19,46,31,34,57,55,10)(5,18,47,30,35,64,56,9)(6,17,48,29,36,63,49,16)(7,24,41,28,37,62,50,15)(8,23,42,27,38,61,51,14), (9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62) );
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),(2,21,44,25,40,59,53,12),(3,20,45,32,33,58,54,11),(4,19,46,31,34,57,55,10),(5,18,47,30,35,64,56,9),(6,17,48,29,36,63,49,16),(7,24,41,28,37,62,50,15),(8,23,42,27,38,61,51,14)], [(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62)])
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 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | Q8 | M4(2) | D8 | Q16 | C8.C4 |
| kernel | C8⋊7M4(2) | C8⋊1C8 | C2×C4×C8 | C4⋊M4(2) | C4×C8 | C22×C8 | C42 | C42 | C22×C4 | C22×C4 | C8 | C2×C4 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 8 | 4 | 4 | 8 |
Matrix representation of C8⋊7M4(2) ►in GL4(𝔽17) generated by
| 8 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 15 | 2 |
| 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 14 | 15 |
| 0 | 0 | 4 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [8,0,0,0,0,15,0,0,0,0,9,15,0,0,0,2],[0,13,0,0,1,0,0,0,0,0,14,4,0,0,15,3],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1] >;
C8⋊7M4(2) in GAP, Magma, Sage, TeX
C_8\rtimes_7M_4(2)
% in TeX
G:=Group("C8:7M4(2)"); // GroupNames label
G:=SmallGroup(128,299);
// by ID
G=gap.SmallGroup(128,299);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,288,1430,1123,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^5>;
// generators/relations