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C22×C5×3−1+2 #10587419

order540
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C2

Aliases

direct productC22×C5×3−1+2

Subgroups

subgroupG/H
center, socleC22×C3×C5C32
commutator, Frattini, cosocleC3C22×C32×C5
prime radicalC22×C32×C5C3
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ G

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×Dic5×3−1+2, 3−1+2×C22⋊D5, C2×C5×3−1+2⋊C4, …
maximal subgroup ofC2×Dic5×3−1+2, 3−1+2×C22⋊D5, C2×C5×3−1+2⋊C4, …
maximal normal subgroup ofC2×Dic5×3−1+2, 3−1+2×C22⋊D5, C2×C5×3−1+2⋊C4, …
maximal quotient ofC2×C4×C5×3−1+2, C5×D4×3−1+2, C5×Q8×3−1+2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(540, 79);