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G = C322C8order 72 = 23·32

The semidirect product of C32 and C8 acting via C8/C2=C4

metabelian, soluble, monomial, A-group

Aliases: C322C8, (C3×C6).C4, C2.(C32⋊C4), C3⋊Dic3.1C2, SmallGroup(72,19)

Series: Derived Chief Lower central Upper central

C1C32 — C322C8
C1C32C3×C6C3⋊Dic3 — C322C8
C32 — C322C8
C1C2

Generators and relations for C322C8
 G = < a,b,c | a3=b3=c8=1, cbc-1=ab=ba, cac-1=a-1b >

2C3
2C3
9C4
2C6
2C6
9C8
6Dic3
6Dic3

Character table of C322C8

 class 123A3B4A4B6A6B8A8B8C8D
 size 114499449999
ρ1111111111111    trivial
ρ211111111-1-1-1-1    linear of order 2
ρ31111-1-111-ii-ii    linear of order 4
ρ41111-1-111i-ii-i    linear of order 4
ρ51-111i-i-1-1ζ87ζ85ζ83ζ8    linear of order 8
ρ61-111-ii-1-1ζ8ζ83ζ85ζ87    linear of order 8
ρ71-111i-i-1-1ζ83ζ8ζ87ζ85    linear of order 8
ρ81-111-ii-1-1ζ85ζ87ζ8ζ83    linear of order 8
ρ9441-200-210000    orthogonal lifted from C32⋊C4
ρ1044-21001-20000    orthogonal lifted from C32⋊C4
ρ114-4-2100-120000    symplectic faithful, Schur index 2
ρ124-41-2002-10000    symplectic faithful, Schur index 2

Permutation representations of C322C8
On 24 points - transitive group 24T63
Generators in S24
(1 21 9)(2 10 22)(3 11 23)(4 24 12)(5 17 13)(6 14 18)(7 15 19)(8 20 16)
(2 22 10)(4 12 24)(6 18 14)(8 16 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

G:=sub<Sym(24)| (1,21,9)(2,10,22)(3,11,23)(4,24,12)(5,17,13)(6,14,18)(7,15,19)(8,20,16), (2,22,10)(4,12,24)(6,18,14)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;

G:=Group( (1,21,9)(2,10,22)(3,11,23)(4,24,12)(5,17,13)(6,14,18)(7,15,19)(8,20,16), (2,22,10)(4,12,24)(6,18,14)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );

G=PermutationGroup([(1,21,9),(2,10,22),(3,11,23),(4,24,12),(5,17,13),(6,14,18),(7,15,19),(8,20,16)], [(2,22,10),(4,12,24),(6,18,14),(8,16,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)])

G:=TransitiveGroup(24,63);

C322C8 is a maximal subgroup of
C2.F9  C32⋊D8  C322SD16  C32⋊Q16  C3⋊S33C8  C32⋊M4(2)  C62.C4  C334C8  (C3×C15)⋊9C8  (C3×C6).F5
C322C8 is a maximal quotient of
C322C16  He32C8  C334C8  (C3×C15)⋊9C8  (C3×C6).F5

Matrix representation of C322C8 in GL4(𝔽5) generated by

4020
0003
2000
0304
,
1000
0402
0010
0200
,
0201
1000
0003
0010
G:=sub<GL(4,GF(5))| [4,0,2,0,0,0,0,3,2,0,0,0,0,3,0,4],[1,0,0,0,0,4,0,2,0,0,1,0,0,2,0,0],[0,1,0,0,2,0,0,0,0,0,0,1,1,0,3,0] >;

C322C8 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2C_8
% in TeX

G:=Group("C3^2:2C8");
// GroupNames label

G:=SmallGroup(72,19);
// by ID

G=gap.SmallGroup(72,19);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,3,10,26,1123,168,1604,609]);
// Polycyclic

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

Export

Subgroup lattice of C322C8 in TeX
Character table of C322C8 in TeX

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