p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.6C42, C42.382D4, C8⋊C4.5C4, C8.C4⋊6C4, C4.174(C4×D4), C22.4(C4×Q8), C4.27(C2×C42), (C22×C4).13Q8, C23.78(C2×Q8), C42.135(C2×C4), (C4×M4(2)).22C2, M4(2).22(C2×C4), C4.C42.8C2, C4.51(C42⋊C2), (C22×C8).211C22, (C2×C42).242C22, C2.3(M4(2).C4), (C22×C4).1319C23, (C2×M4(2)).313C22, C2.17(C4×C4⋊C4), (C2×C8⋊C4).3C2, (C2×C4).81(C4⋊C4), (C2×C8).141(C2×C4), C22.62(C2×C4⋊C4), (C2×C8.C4).9C2, (C2×C4).1510(C2×D4), (C2×C4).550(C4○D4), (C2×C4).526(C22×C4), SmallGroup(128,510)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.6C42
G = < a,b,c | a8=1, b4=c4=a4, bab-1=a3, cac-1=a5, cbc-1=a2b >
Subgroups: 148 in 110 conjugacy classes, 76 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×10], C2×C4 [×2], C2×C4 [×8], C2×C4 [×2], C23, C42 [×4], C2×C8 [×8], C2×C8 [×6], M4(2) [×8], M4(2) [×4], C22×C4, C22×C4 [×2], C4×C8 [×2], C8⋊C4 [×4], C8⋊C4 [×2], C8.C4 [×8], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C4.C42 [×2], C2×C8⋊C4, C4×M4(2) [×2], C2×C8.C4 [×2], C8.6C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×C4⋊C4, M4(2).C4 [×2], C8.6C42
►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 41 7 43 5 45 3 47)(2 44 8 46 6 48 4 42)(9 36 15 38 13 40 11 34)(10 39 16 33 14 35 12 37)(17 26 23 28 21 30 19 32)(18 29 24 31 22 25 20 27)(49 63 55 57 53 59 51 61)(50 58 56 60 54 62 52 64)
(1 52 22 11 5 56 18 15)(2 49 23 16 6 53 19 12)(3 54 24 13 7 50 20 9)(4 51 17 10 8 55 21 14)(25 36 41 58 29 40 45 62)(26 33 42 63 30 37 46 59)(27 38 43 60 31 34 47 64)(28 35 44 57 32 39 48 61)
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), (1,41,7,43,5,45,3,47)(2,44,8,46,6,48,4,42)(9,36,15,38,13,40,11,34)(10,39,16,33,14,35,12,37)(17,26,23,28,21,30,19,32)(18,29,24,31,22,25,20,27)(49,63,55,57,53,59,51,61)(50,58,56,60,54,62,52,64), (1,52,22,11,5,56,18,15)(2,49,23,16,6,53,19,12)(3,54,24,13,7,50,20,9)(4,51,17,10,8,55,21,14)(25,36,41,58,29,40,45,62)(26,33,42,63,30,37,46,59)(27,38,43,60,31,34,47,64)(28,35,44,57,32,39,48,61)>;
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), (1,41,7,43,5,45,3,47)(2,44,8,46,6,48,4,42)(9,36,15,38,13,40,11,34)(10,39,16,33,14,35,12,37)(17,26,23,28,21,30,19,32)(18,29,24,31,22,25,20,27)(49,63,55,57,53,59,51,61)(50,58,56,60,54,62,52,64), (1,52,22,11,5,56,18,15)(2,49,23,16,6,53,19,12)(3,54,24,13,7,50,20,9)(4,51,17,10,8,55,21,14)(25,36,41,58,29,40,45,62)(26,33,42,63,30,37,46,59)(27,38,43,60,31,34,47,64)(28,35,44,57,32,39,48,61) );
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)], [(1,41,7,43,5,45,3,47),(2,44,8,46,6,48,4,42),(9,36,15,38,13,40,11,34),(10,39,16,33,14,35,12,37),(17,26,23,28,21,30,19,32),(18,29,24,31,22,25,20,27),(49,63,55,57,53,59,51,61),(50,58,56,60,54,62,52,64)], [(1,52,22,11,5,56,18,15),(2,49,23,16,6,53,19,12),(3,54,24,13,7,50,20,9),(4,51,17,10,8,55,21,14),(25,36,41,58,29,40,45,62),(26,33,42,63,30,37,46,59),(27,38,43,60,31,34,47,64),(28,35,44,57,32,39,48,61)])
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 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | C4○D4 | M4(2).C4 |
| kernel | C8.6C42 | C4.C42 | C2×C8⋊C4 | C4×M4(2) | C2×C8.C4 | C8⋊C4 | C8.C4 | C42 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C8.6C42 ►in GL6(𝔽17)
| 7 | 16 | 0 | 0 | 0 | 0 |
| 16 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 15 |
| 0 | 0 | 0 | 0 | 15 | 14 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 6 | 4 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 14 | 0 | 0 |
| 0 | 0 | 14 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 12 |
| 0 | 0 | 0 | 0 | 12 | 9 |
G:=sub<GL(6,GF(17))| [7,16,0,0,0,0,16,10,0,0,0,0,0,0,5,8,0,0,0,0,8,12,0,0,0,0,0,0,3,15,0,0,0,0,15,14],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0],[6,4,0,0,0,0,4,11,0,0,0,0,0,0,15,14,0,0,0,0,14,2,0,0,0,0,0,0,8,12,0,0,0,0,12,9] >;
C8.6C42 in GAP, Magma, Sage, TeX
C_8._6C_4^2
% in TeX
G:=Group("C8.6C4^2"); // GroupNames label
G:=SmallGroup(128,510);
// by ID
G=gap.SmallGroup(128,510);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,1018,248,1411,172,124]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^4=c^4=a^4,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^2*b>;
// generators/relations