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G = C8.32D8order 128 = 27

9th non-split extension by C8 of D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D84C8, Q164C8, C8.32D8, C8.29SD16, C8.3M4(2), C42.392D4, C8.3(C2×C8), C8⋊C81C2, C4○D8.3C4, C8○D8.2C2, C8.C87C2, C22.2C4≀C2, (C2×C8).180D4, C2.11(D4⋊C8), C8.C4.2C4, C4.6(C22⋊C8), (C4×C8).131C22, C4.48(D4⋊C4), (C2×C8).48(C2×C4), (C2×C4).216(C22⋊C4), SmallGroup(128,68)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.32D8
C1C2C4C2×C4C2×C8C4×C8C8○D8 — C8.32D8
C1C2C4C8 — C8.32D8
C1C4C2×C8C4×C8 — C8.32D8
C1C2C2C2C2C2×C4C2×C4C4×C8 — C8.32D8

Generators and relations for C8.32D8
 G = < a,b,c | a8=b8=1, c2=a, bab-1=a5, ac=ca, cbc-1=ab-1 >

2C2
8C2
2C4
2C4
4C22
4C4
2D4
2C2×C4
2Q8
4C8
4C2×C4
4C8
4C8
4D4
2C2×C8
2C2×C8
2SD16
2C4○D4
2M4(2)
4C2×C8
4C16
4M4(2)
2C4≀C2
2C8○D4
2C4×C8
2M5(2)

Permutation representations of C8.32D8
On 16 points - transitive group 16T260
Generators in S16
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)
(1 13 9 5)(2 16 6 4 10 8 14 12)(3 7 11 15)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

G:=sub<Sym(16)| (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16), (1,13,9,5)(2,16,6,4,10,8,14,12)(3,7,11,15), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;

G:=Group( (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16), (1,13,9,5)(2,16,6,4,10,8,14,12)(3,7,11,15), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );

G=PermutationGroup([(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16)], [(1,13,9,5),(2,16,6,4,10,8,14,12),(3,7,11,15)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)])

G:=TransitiveGroup(16,260);

32 conjugacy classes

class 1 2A2B2C4A4B4C···4G4H8A8B8C8D8E···8N8O8P16A16B16C16D
order1222444···4488888···88816161616
size1128112···2822224···4888888

32 irreducible representations

dim111111112222224
type+++++++
imageC1C2C2C2C4C4C8C8D4D4M4(2)D8SD16C4≀C2C8.32D8
kernelC8.32D8C8⋊C8C8.C8C8○D8C8.C4C4○D8D8Q16C42C2×C8C8C8C8C22C1
# reps111122441122244

Matrix representation of C8.32D8 in GL4(𝔽5) generated by

0100
2000
0004
0030
,
0021
0022
3031
0332
,
3111
2334
1121
3422
G:=sub<GL(4,GF(5))| [0,2,0,0,1,0,0,0,0,0,0,3,0,0,4,0],[0,0,3,0,0,0,0,3,2,2,3,3,1,2,1,2],[3,2,1,3,1,3,1,4,1,3,2,2,1,4,1,2] >;

C8.32D8 in GAP, Magma, Sage, TeX

C_8._{32}D_8
% in TeX

G:=Group("C8.32D8");
// GroupNames label

G:=SmallGroup(128,68);
// by ID

G=gap.SmallGroup(128,68);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,891,100,1018,136,2804,1411,172,124]);
// Polycyclic

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

Export

Subgroup lattice of C8.32D8 in TeX

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