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G = C8.31D8order 128 = 27

8th non-split extension by C8 of D8 acting via D8/D4=C2

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

Aliases: C8.31D8, C8.36SD16, C4⋊C162C2, C4⋊C4.2C8, C4⋊C8.4C4, C4.41C4≀C2, (C2×D4).3C8, (C4×D4).1C4, C165C46C2, C2.5(D4⋊C8), (C2×C8).371D4, C86D4.10C2, C2.6(D4.C8), C42.42(C2×C4), (C4×C8).303C22, C2.4(C23.C8), (C2×C4).10M4(2), C4.46(D4⋊C4), C22.50(C22⋊C8), (C2×C4).12(C2×C8), (C2×C4).381(C22⋊C4), SmallGroup(128,62)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.31D8
C1C2C4C2×C4C2×C8C4×C8C86D4 — C8.31D8
C1C22C2×C4 — C8.31D8
C1C2×C4C4×C8 — C8.31D8
C1C2C2C2C2C2×C4C2×C4C4×C8 — C8.31D8

Generators and relations for C8.31D8
 G = < a,b,c | a8=1, b8=a4, c2=a, bab-1=a5, ac=ca, cbc-1=a5b7 >

8C2
2C4
2C4
4C4
4C22
4C22
4C22
2C23
2C8
2C2×C4
4D4
4C8
4C2×C4
4C2×C4
2C16
2C16
2C22⋊C4
2C22×C4
2C2×C8
4M4(2)
4M4(2)
4C16
2C2×C16
2C2×C16
2C22⋊C8
2C2×M4(2)

Smallest permutation representation of C8.31D8
On 64 points
Generators in S64
(1 26 55 40 9 18 63 48)(2 19 56 33 10 27 64 41)(3 28 57 42 11 20 49 34)(4 21 58 35 12 29 50 43)(5 30 59 44 13 22 51 36)(6 23 60 37 14 31 52 45)(7 32 61 46 15 24 53 38)(8 25 62 39 16 17 54 47)
(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 17 26 54 55 47 40 8 9 25 18 62 63 39 48 16)(2 61 19 46 56 15 33 24 10 53 27 38 64 7 41 32)(3 37 28 14 57 31 42 52 11 45 20 6 49 23 34 60)(4 5 21 30 58 59 35 44 12 13 29 22 50 51 43 36)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G8A···8H8I8J16A···16P
order1222244444448···88816···16
size1111811114482···2884···4

38 irreducible representations

dim111111112222224
type++++++
imageC1C2C2C2C4C4C8C8D4D8SD16M4(2)C4≀C2D4.C8C23.C8
kernelC8.31D8C165C4C4⋊C16C86D4C4⋊C8C4×D4C4⋊C4C2×D4C2×C8C8C8C2×C4C4C2C2
# reps111122442222482

Matrix representation of C8.31D8 in GL4(𝔽17) generated by

4000
0400
0001
00130
,
01200
111200
00132
0084
,
01200
6000
00132
00913
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,13,0,0,1,0],[0,11,0,0,12,12,0,0,0,0,13,8,0,0,2,4],[0,6,0,0,12,0,0,0,0,0,13,9,0,0,2,13] >;

C8.31D8 in GAP, Magma, Sage, TeX

C_8._{31}D_8
% in TeX

G:=Group("C8.31D8");
// GroupNames label

G:=SmallGroup(128,62);
// by ID

G=gap.SmallGroup(128,62);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,891,436,1018,136,124]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^8=a^4,c^2=a,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^5*b^7>;
// generators/relations

Export

Subgroup lattice of C8.31D8 in TeX

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