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Cayley diagram

C2×C32×3−1+2 #92407

order486
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C2×C34 · Schur multiplier (H2) = C35

Aliases

direct productC2×C32×3−1+2
unambiguous semidirect product(C2×C9)⋊C33

Subgroups

subgroupG/H
center, socleC2×C33C32
commutator, Frattini, cosocleC3C2×C34
prime radicalC2×C34C3
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C2×C33 ◁ G

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC32×3−1+2⋊C4, C3×Dic3×3−1+2, C3×C3⋊(3−1+2⋊C4), …
maximal subgroup ofC32×3−1+2⋊C4, C3×Dic3×3−1+2, C3×C3⋊(3−1+2⋊C4), …
maximal normal subgroup ofC32×3−1+2⋊C4, C3×Dic3×3−1+2, C3×C3⋊(3−1+2⋊C4), …
maximal quotient ofC32×C4×3−1+2, C22×C32×3−1+2, C2×C32×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 252);