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G = C82C8order 64 = 26

2nd semidirect product of C8 and C8 acting via C8/C4=C2

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: C82C8, C4.15SD16, C4.4M4(2), C42.64C22, C4⋊C8.4C2, C4.6(C2×C8), C2.3(C4⋊C8), (C2×C4).8Q8, (C2×C8).11C4, (C4×C8).11C2, (C2×C4).138D4, C2.1(C4.Q8), C2.1(C8.C4), C22.12(C4⋊C4), (C2×C4).59(C2×C4), SmallGroup(64,15)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C82C8
C1C2C22C2×C4C42C4×C8 — C82C8
C1C2C4 — C82C8
C1C2×C4C42 — C82C8
C1C22C22C42 — C82C8

Generators and relations for C82C8
 G = < a,b | a8=b8=1, bab-1=a3 >

2C4
2C8
4C8
4C8
2C2×C8
2C2×C8

Character table of C82C8

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L8M8N8O8P
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1-1-1-1-11-11-11-11    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-11-11-11-11-1    linear of order 2
ρ411111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-1-1-111-1-11-1-1111-ii-iii-ii-i    linear of order 4
ρ61111-1-1-1-1-1-11111-111-1-1-1iiii-i-i-i-i    linear of order 4
ρ71111-1-1-1-1-1-111-1-11-1-1111i-ii-i-ii-ii    linear of order 4
ρ81111-1-1-1-1-1-11111-111-1-1-1-i-i-i-iiiii    linear of order 4
ρ91-11-1i-ii-ii-i1-1-11i1-1-ii-iζ87ζ83ζ83ζ87ζ85ζ8ζ8ζ85    linear of order 8
ρ101-11-1i-ii-ii-i1-1-11i1-1-ii-iζ83ζ87ζ87ζ83ζ8ζ85ζ85ζ8    linear of order 8
ρ111-11-1i-ii-ii-i1-11-1-i-11i-iiζ83ζ83ζ87ζ87ζ8ζ8ζ85ζ85    linear of order 8
ρ121-11-1i-ii-ii-i1-11-1-i-11i-iiζ87ζ87ζ83ζ83ζ85ζ85ζ8ζ8    linear of order 8
ρ131-11-1-ii-ii-ii1-1-11-i1-1i-iiζ8ζ85ζ85ζ8ζ83ζ87ζ87ζ83    linear of order 8
ρ141-11-1-ii-ii-ii1-11-1i-11-ii-iζ85ζ85ζ8ζ8ζ87ζ87ζ83ζ83    linear of order 8
ρ151-11-1-ii-ii-ii1-1-11-i1-1i-iiζ85ζ8ζ8ζ85ζ87ζ83ζ83ζ87    linear of order 8
ρ161-11-1-ii-ii-ii1-11-1i-11-ii-iζ8ζ8ζ85ζ85ζ83ζ83ζ87ζ87    linear of order 8
ρ1722222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ182222-2-2-2-222-2-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ192-22-2-2i2i-2i2i2i-2i-220000000000000000    complex lifted from M4(2)
ρ202-22-22i-2i2i-2i-2i2i-220000000000000000    complex lifted from M4(2)
ρ212-2-22-2i-2i2i2i0000--2--22-2-2-2-2200000000    complex lifted from C8.C4
ρ222-2-222i2i-2i-2i0000-2-22--2--2-2-2200000000    complex lifted from C8.C4
ρ2322-2-2-222-20000-2--2--2-2--2--2-2-200000000    complex lifted from SD16
ρ2422-2-22-2-220000--2-2--2--2-2--2-2-200000000    complex lifted from SD16
ρ2522-2-22-2-220000-2--2-2-2--2-2--2--200000000    complex lifted from SD16
ρ262-2-222i2i-2i-2i0000--2--2-2-2-222-200000000    complex lifted from C8.C4
ρ2722-2-2-222-20000--2-2-2--2-2-2--2--200000000    complex lifted from SD16
ρ282-2-22-2i-2i2i2i0000-2-2-2--2--222-200000000    complex lifted from C8.C4

Smallest permutation representation of C82C8
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 22 55 64 41 25 38 14)(2 17 56 59 42 28 39 9)(3 20 49 62 43 31 40 12)(4 23 50 57 44 26 33 15)(5 18 51 60 45 29 34 10)(6 21 52 63 46 32 35 13)(7 24 53 58 47 27 36 16)(8 19 54 61 48 30 37 11)

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

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

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

C82C8 is a maximal subgroup of
D8⋊C8  Q16⋊C8  C42.42Q8  M4(2)⋊1C8  C88M4(2)  C42.43Q8  C81M4(2)  C42.90D4  C42.Q8  C42.92D4  C42.21Q8  C8×SD16  Q165C8  D85C8  C812SD16  C815SD16  D42M4(2)  Q8.M4(2)  C89SD16  C8.M4(2)  C83M4(2)  C88D8  C814SD16  C811SD16  C88Q16  D4.2SD16  Q8.2SD16  D4.3SD16  Q8.3SD16  C82D8  C82SD16  C8.D8  C84SD16  C82Q16  C8.3Q16  C42.252C23  C42.253C23  C42.254C23  C42.255C23  C20.26M4(2)
 C8p⋊C8: C161C8  C242C8  C406C8  C402C8  C562C8 ...
 C4p.SD16: C8.30D8  C8.16Q16  C8.SD16  C8.8SD16  C12.39SD16  C20.39SD16  C28.39SD16 ...
C82C8 is a maximal quotient of
C42.385D4  C20.26M4(2)
 C8p⋊C8: C161C8  C242C8  C406C8  C402C8  C562C8 ...
 C4p.SD16: C82C16  C16.C8  C12.39SD16  C20.39SD16  C28.39SD16 ...

Matrix representation of C82C8 in GL3(𝔽17) generated by

100
0010
01210
,
200
0012
060
G:=sub<GL(3,GF(17))| [1,0,0,0,0,12,0,10,10],[2,0,0,0,0,6,0,12,0] >;

C82C8 in GAP, Magma, Sage, TeX

C_8\rtimes_2C_8
% in TeX

G:=Group("C8:2C8");
// GroupNames label

G:=SmallGroup(64,15);
// by ID

G=gap.SmallGroup(64,15);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,31,362,86,117]);
// Polycyclic

G:=Group<a,b|a^8=b^8=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C82C8 in TeX
Character table of C82C8 in TeX

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