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C3×3−1+2×#442 #72254692

order5184
nilpotency class3
automorphism?
homologyAbelianization (H1) = C22×C33×C4 · Schur multiplier (H2) = C23×C32

Aliases

direct productC3×3−1+2×#442

Subgroups

subgroupG/H
center, socleC22×C32C2×C32×D4
commutatorC3×C4C22×C33×C4
Frattini, cosocleC2×C3×C4C23×C33
prime radicalC2×C33×D8C2×C3
LCSG ▷ C3×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C32 ◁ C3×C42×3−1+2 ◁ G

2508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 864

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of#442×C3⋊AGL(1,ℤ/9)
maximal subgroup of#442×C3⋊AGL(1,ℤ/9)
maximal normal subgroup of#442×C3⋊AGL(1,ℤ/9)

GAP

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

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