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G = C8⋊C8order 64 = 26

3rd semidirect product of C8 and C8 acting via C8/C4=C2

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: C83C8, C4.9M4(2), C22.6C42, C42.91C22, (C2×C8).4C4, C2.1(C4×C8), C4.11(C2×C8), (C4×C8).13C2, C2.1(C8⋊C4), (C2×C4).78(C2×C4), SmallGroup(64,3)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8⋊C8
C1C2C22C2×C4C42C4×C8 — C8⋊C8
C1C2 — C8⋊C8
C1C42 — C8⋊C8
C1C22C22C42 — C8⋊C8

Generators and relations for C8⋊C8
 G = < a,b | a8=b8=1, bab-1=a5 >

2C8
2C8
2C8
2C8

Smallest permutation representation of C8⋊C8
Regular action on 64 points
Generators in S64
(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 29 20 46 37 59 56 15)(2 26 21 43 38 64 49 12)(3 31 22 48 39 61 50 9)(4 28 23 45 40 58 51 14)(5 25 24 42 33 63 52 11)(6 30 17 47 34 60 53 16)(7 27 18 44 35 57 54 13)(8 32 19 41 36 62 55 10)

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), (1,29,20,46,37,59,56,15)(2,26,21,43,38,64,49,12)(3,31,22,48,39,61,50,9)(4,28,23,45,40,58,51,14)(5,25,24,42,33,63,52,11)(6,30,17,47,34,60,53,16)(7,27,18,44,35,57,54,13)(8,32,19,41,36,62,55,10)>;

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), (1,29,20,46,37,59,56,15)(2,26,21,43,38,64,49,12)(3,31,22,48,39,61,50,9)(4,28,23,45,40,58,51,14)(5,25,24,42,33,63,52,11)(6,30,17,47,34,60,53,16)(7,27,18,44,35,57,54,13)(8,32,19,41,36,62,55,10) );

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)], [(1,29,20,46,37,59,56,15),(2,26,21,43,38,64,49,12),(3,31,22,48,39,61,50,9),(4,28,23,45,40,58,51,14),(5,25,24,42,33,63,52,11),(6,30,17,47,34,60,53,16),(7,27,18,44,35,57,54,13),(8,32,19,41,36,62,55,10)])

C8⋊C8 is a maximal subgroup of
C8×M4(2)  C82⋊C2  C89M4(2)  C23.27C42  C822C2  C86M4(2)  SD16⋊C8  Q165C8  D85C8  C89D8  C812SD16  C815SD16  C89Q16  D4.M4(2)  D42M4(2)  Q8.M4(2)  Q82M4(2)  C8⋊M4(2)  C83M4(2)  C85SD16  C86SD16  C8.9SD16  C42.664C23  C42.665C23  C42.666C23  C42.667C23  C83D8  C8.2D8  C83Q16
 C8p⋊C8: C16⋊C8  C24⋊C8  C408C8  C40⋊C8  C56⋊C8 ...
 C4p.M4(2): C8.32D8  C8215C2  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  C408C8  C40⋊C8  C56⋊C8 ...
 C42.D2p: C2.C82  C42.279D6  C42.279D10  C42.279D14 ...

40 conjugacy classes

class 1 2A2B2C4A···4L8A···8X
order12224···48···8
size11111···12···2

40 irreducible representations

dim11112
type++
imageC1C2C4C8M4(2)
kernelC8⋊C8C4×C8C2×C8C8C4
# reps1312168

Matrix representation of C8⋊C8 in GL3(𝔽17) generated by

100
0515
0612
,
200
010
0516
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

Export

Subgroup lattice of C8⋊C8 in TeX

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