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, C2, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C23, C16, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C8.C4, C8.C4, C2×C16, C2×C16, C22×C8, C2×M4(2), C2×C8.C4, C22×C16, C8.8C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C22.4Q16, C8.4Q8, C8.8C42
►On 64 points
(1 24 13 20 9 32 5 28)(2 25 14 21 10 17 6 29)(3 26 15 22 11 18 7 30)(4 27 16 23 12 19 8 31)(33 49 45 61 41 57 37 53)(34 50 46 62 42 58 38 54)(35 51 47 63 43 59 39 55)(36 52 48 64 44 60 40 56)
(1 62 13 50 9 54 5 58)(2 43 14 47 10 35 6 39)(3 60 15 64 11 52 7 56)(4 41 16 45 12 33 8 37)(17 55 29 59 25 63 21 51)(18 36 30 40 26 44 22 48)(19 53 31 57 27 61 23 49)(20 34 32 38 28 42 24 46)
(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)| (1,24,13,20,9,32,5,28)(2,25,14,21,10,17,6,29)(3,26,15,22,11,18,7,30)(4,27,16,23,12,19,8,31)(33,49,45,61,41,57,37,53)(34,50,46,62,42,58,38,54)(35,51,47,63,43,59,39,55)(36,52,48,64,44,60,40,56), (1,62,13,50,9,54,5,58)(2,43,14,47,10,35,6,39)(3,60,15,64,11,52,7,56)(4,41,16,45,12,33,8,37)(17,55,29,59,25,63,21,51)(18,36,30,40,26,44,22,48)(19,53,31,57,27,61,23,49)(20,34,32,38,28,42,24,46), (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( (1,24,13,20,9,32,5,28)(2,25,14,21,10,17,6,29)(3,26,15,22,11,18,7,30)(4,27,16,23,12,19,8,31)(33,49,45,61,41,57,37,53)(34,50,46,62,42,58,38,54)(35,51,47,63,43,59,39,55)(36,52,48,64,44,60,40,56), (1,62,13,50,9,54,5,58)(2,43,14,47,10,35,6,39)(3,60,15,64,11,52,7,56)(4,41,16,45,12,33,8,37)(17,55,29,59,25,63,21,51)(18,36,30,40,26,44,22,48)(19,53,31,57,27,61,23,49)(20,34,32,38,28,42,24,46), (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([[(1,24,13,20,9,32,5,28),(2,25,14,21,10,17,6,29),(3,26,15,22,11,18,7,30),(4,27,16,23,12,19,8,31),(33,49,45,61,41,57,37,53),(34,50,46,62,42,58,38,54),(35,51,47,63,43,59,39,55),(36,52,48,64,44,60,40,56)], [(1,62,13,50,9,54,5,58),(2,43,14,47,10,35,6,39),(3,60,15,64,11,52,7,56),(4,41,16,45,12,33,8,37),(17,55,29,59,25,63,21,51),(18,36,30,40,26,44,22,48),(19,53,31,57,27,61,23,49),(20,34,32,38,28,42,24,46)], [(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