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G = C8.C22order 32 = 25

The non-split extension by C8 of C22 acting faithfully

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

Aliases: C8.C22, Q162C2, C4.15D4, SD162C2, C4.6C23, C22.6D4, M4(2)⋊2C2, D4.3C22, Q8.3C22, (C2×Q8)⋊4C2, C4○D4.2C2, C2.16(C2×D4), (C2×C4).7C22, 2-Sylow(ASigmaL(2,9)), SmallGroup(32,44)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.C22
C1C2C4C2×C4C2×Q8 — 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, cbc=a4b >

2C2
4C2
2C4
2C22
2C4
2C4
2D4
2Q8
2C2×C4
2C2×C4

Character table of C8.C22

 class 12A2B2C4A4B4C4D4E8A8B
 size 11242244444
ρ111111111111    trivial
ρ2111-1111-11-1-1    linear of order 2
ρ311-1-11-1-111-11    linear of order 2
ρ411-111-1-1-111-1    linear of order 2
ρ511-111-11-1-1-11    linear of order 2
ρ611-1-11-111-11-1    linear of order 2
ρ7111-111-1-1-111    linear of order 2
ρ8111111-11-1-1-1    linear of order 2
ρ92220-2-200000    orthogonal lifted from D4
ρ1022-20-2200000    orthogonal lifted from D4
ρ114-4000000000    symplectic faithful, Schur index 2

Permutation representations of C8.C22
On 16 points - transitive group 16T32
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)

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

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

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

G:=TransitiveGroup(16,32);

On 16 points - transitive group 16T50
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 15)(3 10)(4 13)(5 16)(6 11)(7 14)(8 9)
(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), (1,12)(2,15)(3,10)(4,13)(5,16)(6,11)(7,14)(8,9), (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), (1,12)(2,15)(3,10)(4,13)(5,16)(6,11)(7,14)(8,9), (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)], [(1,12),(2,15),(3,10),(4,13),(5,16),(6,11),(7,14),(8,9)], [(2,6),(4,8),(10,14),(12,16)])

G:=TransitiveGroup(16,50);

Matrix representation of C8.C22 in GL4(𝔽3) generated by

0200
1010
0202
0010
,
0202
0010
0100
2020
,
2000
0100
0020
0001
G:=sub<GL(4,GF(3))| [0,1,0,0,2,0,2,0,0,1,0,1,0,0,2,0],[0,0,0,2,2,0,1,0,0,1,0,2,2,0,0,0],[2,0,0,0,0,1,0,0,0,0,2,0,0,0,0,1] >;

C8.C22 in GAP, Magma, Sage, TeX

C_8.C_2^2
% in TeX

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

G:=SmallGroup(32,44);
// by ID

G=gap.SmallGroup(32,44);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,101,86,302,483,248,58]);
// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^2=1,b*a*b=a^3,c*a*c=a^5,c*b*c=a^4*b>;
// generators/relations

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