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G = C22⋊C8order 32 = 25

The semidirect product of C22 and C8 acting via C8/C4=C2

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

Aliases: C22⋊C8, C4.16D4, C23.2C4, C2.2M4(2), (C2×C8)⋊1C2, (C2×C4).3C4, C2.1(C2×C8), (C22×C4).2C2, C22.8(C2×C4), C2.2(C22⋊C4), (C2×C4).32C22, SmallGroup(32,5)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22⋊C8
C1C2C4C2×C4C22×C4 — C22⋊C8
C1C2 — C22⋊C8
C1C2×C4 — C22⋊C8
C1C2C2C2×C4 — C22⋊C8

Generators and relations for C22⋊C8
 G = < a,b,c | a2=b2=c8=1, cac-1=ab=ba, bc=cb >

2C2
2C2
2C22
2C4
2C22
2C8
2C2×C4
2C2×C4
2C8

Character table of C22⋊C8

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F8G8H
 size 11112211112222222222
ρ111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-11111-1-1-1-1-1-11111    linear of order 2
ρ41111-1-11111-1-11111-1-1-1-1    linear of order 2
ρ5111111-1-1-1-1-1-1i-ii-i-iii-i    linear of order 4
ρ6111111-1-1-1-1-1-1-ii-iii-i-ii    linear of order 4
ρ71111-1-1-1-1-1-111-ii-ii-iii-i    linear of order 4
ρ81111-1-1-1-1-1-111i-ii-ii-i-ii    linear of order 4
ρ91-11-11-1ii-i-i-iiζ83ζ8ζ87ζ85ζ8ζ83ζ87ζ85    linear of order 8
ρ101-11-11-1-i-iiii-iζ8ζ83ζ85ζ87ζ83ζ8ζ85ζ87    linear of order 8
ρ111-11-11-1ii-i-i-iiζ87ζ85ζ83ζ8ζ85ζ87ζ83ζ8    linear of order 8
ρ121-11-11-1-i-iiii-iζ85ζ87ζ8ζ83ζ87ζ85ζ8ζ83    linear of order 8
ρ131-11-1-11-i-iii-iiζ85ζ87ζ8ζ83ζ83ζ8ζ85ζ87    linear of order 8
ρ141-11-1-11ii-i-ii-iζ83ζ8ζ87ζ85ζ85ζ87ζ83ζ8    linear of order 8
ρ151-11-1-11-i-iii-iiζ8ζ83ζ85ζ87ζ87ζ85ζ8ζ83    linear of order 8
ρ161-11-1-11ii-i-ii-iζ87ζ85ζ83ζ8ζ8ζ83ζ87ζ85    linear of order 8
ρ1722-2-200-22-220000000000    orthogonal lifted from D4
ρ1822-2-2002-22-20000000000    orthogonal lifted from D4
ρ192-2-2200-2i2i2i-2i0000000000    complex lifted from M4(2)
ρ202-2-22002i-2i-2i2i0000000000    complex lifted from M4(2)

Permutation representations of C22⋊C8
On 16 points - transitive group 16T24
Generators in S16
(2 12)(4 14)(6 16)(8 10)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,24);

Matrix representation of C22⋊C8 in GL3(𝔽17) generated by

100
0160
001
,
100
0160
0016
,
1500
001
040
G:=sub<GL(3,GF(17))| [1,0,0,0,16,0,0,0,1],[1,0,0,0,16,0,0,0,16],[15,0,0,0,0,4,0,1,0] >;

C22⋊C8 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_8
% in TeX

G:=Group("C2^2:C8");
// GroupNames label

G:=SmallGroup(32,5);
// by ID

G=gap.SmallGroup(32,5);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,58]);
// Polycyclic

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

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