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G = C8≀C2order 128 = 27

Wreath product of C8 by C2

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

Aliases: C8C2, D82C8, C828C2, Q162C8, C8.35D8, C8.23SD16, C8.9M4(2), C42.391D4, C8.9(C2×C8), C4○D8.1C4, C8○D8.1C2, C8.C86C2, C22.1C4≀C2, (C2×C8).282D4, C2.10(D4⋊C8), C8.C4.1C4, C4.5(C22⋊C8), (C4×C8).418C22, C4.47(D4⋊C4), (C2×C8).177(C2×C4), (C2×C4).215(C22⋊C4), 2-Sylow(GL(2,9)), SmallGroup(128,67)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8≀C2
C1C2C4C2×C4C2×C8C4×C8C8○D8 — C8≀C2
C1C2C4C8 — C8≀C2
C1C8C2×C8C4×C8 — C8≀C2
C1C2C2C2C2C2×C4C2×C4C4×C8 — C8≀C2

Generators and relations for C8≀C2
 G = < a,b,c | a8=b2=c8=1, bab=a-1, ac=ca, cbc-1=a-1b >

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

Permutation representations of C8≀C2
On 16 points - transitive group 16T289
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)
(1 3 5 7)(2 4 6 8)(9 14 11 16 13 10 15 12)

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

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

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

G:=TransitiveGroup(16,289);

44 conjugacy classes

class 1 2A2B2C4A4B4C···4G4H8A8B8C8D8E···8Z8AA8AB16A16B16C16D
order1222444···4488888···88816161616
size1128112···2811112···2888888

44 irreducible representations

dim111111112222222
type+++++++
imageC1C2C2C2C4C4C8C8D4D4M4(2)D8SD16C4≀C2C8≀C2
kernelC8≀C2C82C8.C8C8○D8C8.C4C4○D8D8Q16C42C2×C8C8C8C8C22C1
# reps1111224411222416

Matrix representation of C8≀C2 in GL2(𝔽17) generated by

150
08
,
08
150
,
130
08
G:=sub<GL(2,GF(17))| [15,0,0,8],[0,15,8,0],[13,0,0,8] >;

C8≀C2 in GAP, Magma, Sage, TeX

C_8\wr C_2
% in TeX

G:=Group("C8wrC2");
// GroupNames label

G:=SmallGroup(128,67);
// by ID

G=gap.SmallGroup(128,67);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,219,436,136,2804,1411,172,124]);
// Polycyclic

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

Export

Subgroup lattice of C8≀C2 in TeX

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