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G = C8⋊C22order 32 = 25

The semidirect product of C8 and C22 acting faithfully

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

Aliases: C8⋊C22, D82C2, C4.14D4, SD161C2, D42C22, C4.5C23, Q82C22, C22.5D4, M4(2)⋊1C2, C4○D42C2, (C2×D4)⋊5C2, C2.15(C2×D4), (C2×C4).6C22, 2-Sylow(PGammaL(2,9)), Aut(D8), Hol(C8), SmallGroup(32,43)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8⋊C22
C1C2C4C2×C4C2×D4 — C8⋊C22
C1C2C4 — C8⋊C22
C1C2C2×C4 — C8⋊C22
C1C2C2C4 — C8⋊C22

Generators and relations for C8⋊C22
 G = < a,b,c | a8=b2=c2=1, bab=a3, cac=a5, bc=cb >

2C2
4C2
4C2
4C2
2C4
2C22
2C22
2C22
4C22
4C22
2D4
2D4
2C2×C4
2C23

Character table of C8⋊C22

 class 12A2B2C2D2E4A4B4C8A8B
 size 11244422444
ρ111111111111    trivial
ρ2111-11-1111-1-1    linear of order 2
ρ311-11-1-11-111-1    linear of order 2
ρ411-1-1-111-11-11    linear of order 2
ρ511-111-11-1-1-11    linear of order 2
ρ611-1-1111-1-11-1    linear of order 2
ρ71111-1111-1-1-1    linear of order 2
ρ8111-1-1-111-111    linear of order 2
ρ9222000-2-2000    orthogonal lifted from D4
ρ1022-2000-22000    orthogonal lifted from D4
ρ114-4000000000    orthogonal faithful

Permutation representations of C8⋊C22
On 8 points - transitive group 8T15
Generators in S8
(1 2 3 4 5 6 7 8)
(2 4)(3 7)(6 8)
(1 5)(3 7)

G:=sub<Sym(8)| (1,2,3,4,5,6,7,8), (2,4)(3,7)(6,8), (1,5)(3,7)>;

G:=Group( (1,2,3,4,5,6,7,8), (2,4)(3,7)(6,8), (1,5)(3,7) );

G=PermutationGroup([(1,2,3,4,5,6,7,8)], [(2,4),(3,7),(6,8)], [(1,5),(3,7)])

G:=TransitiveGroup(8,15);

On 16 points - transitive group 16T35
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9)(2 12)(3 15)(4 10)(5 13)(6 16)(7 11)(8 14)
(2 6)(4 8)(10 14)(12 16)

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

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

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

G:=TransitiveGroup(16,35);

On 16 points - transitive group 16T38
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 9)(3 12)(4 15)(5 10)(6 13)(7 16)(8 11)
(1 14)(2 11)(3 16)(4 13)(5 10)(6 15)(7 12)(8 9)

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

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

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

G:=TransitiveGroup(16,38);

On 16 points - transitive group 16T45
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)
(1 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)

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

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

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

G:=TransitiveGroup(16,45);

Polynomial with Galois group C8⋊C22 over ℚ
actionf(x)Disc(f)
8T15x8-x4-1-216·54

Matrix representation of C8⋊C22 in GL4(ℤ) generated by

0010
000-1
0100
1000
,
1000
0-100
000-1
00-10
,
1000
0100
00-10
000-1
G:=sub<GL(4,Integers())| [0,0,0,1,0,0,1,0,1,0,0,0,0,-1,0,0],[1,0,0,0,0,-1,0,0,0,0,0,-1,0,0,-1,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1] >;

C8⋊C22 in GAP, Magma, Sage, TeX

C_8\rtimes C_2^2
% in TeX

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

G:=SmallGroup(32,43);
// by ID

G=gap.SmallGroup(32,43);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,101,302,483,248,58]);
// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^2=1,b*a*b=a^3,c*a*c=a^5,b*c=c*b>;
// generators/relations

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