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Cayley diagram

C2×#731 #4193

order162
nilpotency class3
isoclinismisoclinic class of AΣL(1,𝔽27)
automorphismAut(G) = C3×((C3⋊D3)⋊C3)⋊C2 (order 324) · Inn(G) = H(3,𝔽3) · Out(G) = C22×C3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC2×#731

Subgroups

subgroupG/H
center, socleC2×C3H(3,𝔽3)
commutator, Frattini, cosocleC32C2×C32
prime radicalC2×H(3,𝔽3)C3
LCSG ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C32 ◁ G

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2)(4,7,13)(5,10,20)(8,17,27)(9,16,24)(11,22,21)(14,18,23)(15,19,26)(25,28,29);(3,4,8,6,9,18,12,19,28)(5,7,14,10,16,25,20,26,27)(11,15,17,21,13,23,22,24,29)

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10619556, #10619557, #10619558, …
Fitting subgroup of#66323, #66324, #66331, …
maximal subgroup of#66323, #66324, #66331, …
maximal normal subgroup of#66323, #66324, #66331, …
maximal quotient ofC4×#731, C22×#731, C2×#6857, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 29);