p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C8:3C8, C4.9M4(2), C22.6C42, C42.91C22, (C2xC8).4C4, C2.1(C4xC8), C4.11(C2xC8), (C4xC8).13C2, C2.1(C8:C4), (C2xC4).78(C2xC4), SmallGroup(64,3)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8:C8
G = < a,b | a8=b8=1, bab-1=a5 >
Quotients: C1, C2, C4, C22, C8, C2xC4, C42, C2xC8, M4(2), C4xC8, C8:C4, C8:C8
Smallest permutation representation of C8:C8
►Regular action 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 63 17 46 39 25 56 9)(2 60 18 43 40 30 49 14)(3 57 19 48 33 27 50 11)(4 62 20 45 34 32 51 16)(5 59 21 42 35 29 52 13)(6 64 22 47 36 26 53 10)(7 61 23 44 37 31 54 15)(8 58 24 41 38 28 55 12)
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), (1,63,17,46,39,25,56,9)(2,60,18,43,40,30,49,14)(3,57,19,48,33,27,50,11)(4,62,20,45,34,32,51,16)(5,59,21,42,35,29,52,13)(6,64,22,47,36,26,53,10)(7,61,23,44,37,31,54,15)(8,58,24,41,38,28,55,12)>;
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), (1,63,17,46,39,25,56,9)(2,60,18,43,40,30,49,14)(3,57,19,48,33,27,50,11)(4,62,20,45,34,32,51,16)(5,59,21,42,35,29,52,13)(6,64,22,47,36,26,53,10)(7,61,23,44,37,31,54,15)(8,58,24,41,38,28,55,12) );
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)], [(1,63,17,46,39,25,56,9),(2,60,18,43,40,30,49,14),(3,57,19,48,33,27,50,11),(4,62,20,45,34,32,51,16),(5,59,21,42,35,29,52,13),(6,64,22,47,36,26,53,10),(7,61,23,44,37,31,54,15),(8,58,24,41,38,28,55,12)]])
C8:C8 is a maximal subgroup of
C8xM4(2) C82:C2 C8:9M4(2) C23.27C42 C82:2C2 C8:6M4(2) SD16:C8 Q16:5C8 D8:5C8 C8:9D8 C8:12SD16 C8:15SD16 C8:9Q16 D4.M4(2) D4:2M4(2) Q8.M4(2) Q8:2M4(2) C8:M4(2) C8:3M4(2) C8:5SD16 C8:6SD16 C8.9SD16 C42.664C23 C42.665C23 C42.666C23 C42.667C23 C8:3D8 C8.2D8 C8:3Q16
C8p:C8: C16:C8 C24:C8 C40:8C8 C40:C8 C56:C8 ...
C4p.M4(2): C8.32D8 C82:15C2 C8.M4(2) C42.279D6 C42.279D10 C20.31M4(2) C42.279D14 ...
C8:C8 is a maximal quotient of
C8:C16 C20.31M4(2)
C8p:C8: C16:C8 C24:C8 C40:8C8 C40:C8 C56:C8 ...
C42.D2p: C2.C82 C42.279D6 C42.279D10 C42.279D14 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 |
| type | + | + | |||
| image | C1 | C2 | C4 | C8 | M4(2) |
| kernel | C8:C8 | C4xC8 | C2xC8 | C8 | C4 |
| # reps | 1 | 3 | 12 | 16 | 8 |
Matrix representation of C8:C8 ►in GL3(F17) generated by
| 1 | 0 | 0 |
| 0 | 5 | 15 |
| 0 | 6 | 12 |
| 2 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 5 | 16 |
G:=sub<GL(3,GF(17))| [1,0,0,0,5,6,0,15,12],[2,0,0,0,1,5,0,0,16] >;
C8:C8 in GAP, Magma, Sage, TeX
C_8\rtimes C_8
% in TeX
G:=Group("C8:C8"); // GroupNames label
G:=SmallGroup(64,3);
// by ID
G=gap.SmallGroup(64,3);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,24,217,55,86,117]);
// Polycyclic
G:=Group<a,b|a^8=b^8=1,b*a*b^-1=a^5>;
// generators/relations
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