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C22×C4⋊(C3⋊AGL(1,ℤ/9)) #72268902

order2592
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C3

Aliases

direct productC22×C4⋊(C3⋊AGL(1,ℤ/9))

Splittings

FittingC22×C3×C4×3−1+2 ⋊ C2
SylowC3×3−1+2 ⋊ C22×D4

Subgroups

subgroupG/H
centerC23C2×C3⋊AGL(1,ℤ/9)
commutator, cosocleC2×C3×C9C24×C3
FrattiniC2×C3C23×C3×C3⋊D3
FittingC22×C3×C4×3−1+2C2
socleC23×C32C2×C3×D3
hypercenterC22×C4C3⋊AGL(1,ℤ/9)
hypocenterC3×C9C22×C3×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

2603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 432

A minimal faithful permutation representation — 4 generators on 20 points:

(5,6,7)(8,9,10,11)(13,16,18)(15,17,20);(6,7)(9,11)(13,15)(14,19)(16,17)(18,20);(3,4)(12,16,20,14,13,15,19,18,17);(1,2)

Conjugacy classes & character table

300 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC22×D4×C3⋊AGL(1,ℤ/9), C22×C4⋊#92314
maximal normal subgroup ofC22×D4×C3⋊AGL(1,ℤ/9)
maximal quotient ofC22×C4⋊#92314

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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