p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.8C42, C23.11Q16, (C2×C16).8C4, C8.26(C4⋊C4), (C2×C8).52Q8, (C2×C8).353D4, (C2×C4).162D8, C4.9(C4.Q8), C8.C4.3C4, (C22×C16).4C2, (C2×C4).65SD16, C8.34(C22⋊C4), (C22×C4).570D4, C2.3(C8.4Q8), C4.49(D4⋊C4), C22.19(C2.D8), C4.2(C2.C42), (C22×C8).545C22, C22.8(Q8⋊C4), C2.11(C22.4Q16), (C2×C8).178(C2×C4), (C2×C4).109(C4⋊C4), (C2×C8.C4).1C2, (C2×C4).228(C22⋊C4), SmallGroup(128,113)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.8C42
G = < a,b,c | a8=1, b4=a4, c4=a6, bab-1=a-1, ac=ca, cbc-1=a-1b >
Subgroups: 104 in 64 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×2], C8 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C16 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×6], C22×C4, C8.C4 [×4], C8.C4 [×2], C2×C16 [×2], C2×C16 [×2], C22×C8, C2×M4(2) [×2], C2×C8.C4 [×2], C22×C16, C8.8C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C22.4Q16, C8.4Q8 [×2], C8.8C42
►On 64 points
(1 52 13 64 9 60 5 56)(2 53 14 49 10 61 6 57)(3 54 15 50 11 62 7 58)(4 55 16 51 12 63 8 59)(17 33 29 45 25 41 21 37)(18 34 30 46 26 42 22 38)(19 35 31 47 27 43 23 39)(20 36 32 48 28 44 24 40)
(1 44 13 48 9 36 5 40)(2 25 14 29 10 17 6 21)(3 42 15 46 11 34 7 38)(4 23 16 27 12 31 8 19)(18 58 22 54 26 50 30 62)(20 56 24 52 28 64 32 60)(33 61 37 57 41 53 45 49)(35 59 39 55 43 51 47 63)
(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)
G:=sub<Sym(64)| (1,52,13,64,9,60,5,56)(2,53,14,49,10,61,6,57)(3,54,15,50,11,62,7,58)(4,55,16,51,12,63,8,59)(17,33,29,45,25,41,21,37)(18,34,30,46,26,42,22,38)(19,35,31,47,27,43,23,39)(20,36,32,48,28,44,24,40), (1,44,13,48,9,36,5,40)(2,25,14,29,10,17,6,21)(3,42,15,46,11,34,7,38)(4,23,16,27,12,31,8,19)(18,58,22,54,26,50,30,62)(20,56,24,52,28,64,32,60)(33,61,37,57,41,53,45,49)(35,59,39,55,43,51,47,63), (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)>;
G:=Group( (1,52,13,64,9,60,5,56)(2,53,14,49,10,61,6,57)(3,54,15,50,11,62,7,58)(4,55,16,51,12,63,8,59)(17,33,29,45,25,41,21,37)(18,34,30,46,26,42,22,38)(19,35,31,47,27,43,23,39)(20,36,32,48,28,44,24,40), (1,44,13,48,9,36,5,40)(2,25,14,29,10,17,6,21)(3,42,15,46,11,34,7,38)(4,23,16,27,12,31,8,19)(18,58,22,54,26,50,30,62)(20,56,24,52,28,64,32,60)(33,61,37,57,41,53,45,49)(35,59,39,55,43,51,47,63), (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) );
G=PermutationGroup([(1,52,13,64,9,60,5,56),(2,53,14,49,10,61,6,57),(3,54,15,50,11,62,7,58),(4,55,16,51,12,63,8,59),(17,33,29,45,25,41,21,37),(18,34,30,46,26,42,22,38),(19,35,31,47,27,43,23,39),(20,36,32,48,28,44,24,40)], [(1,44,13,48,9,36,5,40),(2,25,14,29,10,17,6,21),(3,42,15,46,11,34,7,38),(4,23,16,27,12,31,8,19),(18,58,22,54,26,50,30,62),(20,56,24,52,28,64,32,60),(33,61,37,57,41,53,45,49),(35,59,39,55,43,51,47,63)], [(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)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 8I | ··· | 8P | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | C8.4Q8 |
| kernel | C8.8C42 | C2×C8.C4 | C22×C16 | C8.C4 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 16 |
Matrix representation of C8.8C42 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 9 | 13 |
| 0 | 0 | 2 |
| 13 | 0 | 0 |
| 0 | 7 | 4 |
| 0 | 8 | 10 |
| 13 | 0 | 0 |
| 0 | 11 | 9 |
| 0 | 0 | 14 |
G:=sub<GL(3,GF(17))| [1,0,0,0,9,0,0,13,2],[13,0,0,0,7,8,0,4,10],[13,0,0,0,11,0,0,9,14] >;
C8.8C42 in GAP, Magma, Sage, TeX
C_8._8C_4^2
% in TeX
G:=Group("C8.8C4^2"); // GroupNames label
G:=SmallGroup(128,113);
// by ID
G=gap.SmallGroup(128,113);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,520,248,3924,102,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^4=a^4,c^4=a^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b>;
// generators/relations