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G = C22⋊C8  order 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

Derived series
C1 — C2 — C22⋊C8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22⋊C8
Lower central
C1 — C2 — C22⋊C8
Upper central
C1 — C2×C4 — C22⋊C8
Jennings
C1 — C2 — C2 — C2×C4 — C22⋊C8

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

C1
C2
C2
C2
2C2
2C2
C4
C22
C22
C4
C22
2C22
2C4
2C22
C2×C4
C2×C4
C23
2C8
2C2×C4
2C2×C4
2C8
C2×C8
C2×C8
C22×C4
C22⋊C8

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);
 

C22⋊C8 is a maximal subgroup of
 C23⋊C8  C22.M4(2)  C22.SD16  C23.31D4  C24.4C4  (C22×C8)⋊C2  C42.7C22  C8×D4  C8⋊9D4  C8⋊6D4  C22⋊D8  Q8⋊D4  D4⋊D4  C22⋊SD16  C22⋊Q16  D4.7D4  C22.D8  C23.46D4  C23.19D4  C23.47D4  C23.48D4  C23.20D4  A4⋊C8  S32⋊C8  C62.6(C2×C4)  C62⋊3C8  C22⋊F9
 C2p.M4(2): C42.12C4  C42.6C4  D6⋊C8  C12.55D4  D10⋊1C8  C20.55D4  D10⋊C8  C23.2F5 ...
C22⋊C8 is a maximal quotient of
 C23⋊C8  C22.M4(2)  Q8⋊C8  C22.7C42  C23.2F5  S32⋊C8  C62.6(C2×C4)  C62⋊3C8  C22⋊F9  C26.M4(2)
 D2p⋊C8: D4⋊C8  D6⋊C8  D10⋊1C8  D10⋊C8  D14⋊C8  D22⋊C8  D26⋊1C8  D26⋊C8 ...
 C4p.D4: C22⋊C16  C23.C8  D4.C8  C12.55D4  C20.55D4  C28.55D4  C44.55D4  C52.55D4 ...

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
 

Export

Subgroup lattice of C22⋊C8 in TeX
Character table of C22⋊C8 in TeX

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