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C27×3−1+2 #10624726

order729
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C32×C27 · Schur multiplier (H2) = C32

Aliases

direct productC27×3−1+2

Subgroups

subgroupG/H
centerC3×C27C32
commutatorC3C32×C27
Frattini, cosocleC3×C9C33
socleC32C32×C9
prime radicalC33C3×C9
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C3×C27 ◁ G
LECSG ▷ C3×C9 ▷ C3 ▷ C1

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 243

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27);(28,29,31,30,33,35,34,32,36);(29,32,33)(31,35,36)

Conjugacy classes & character table

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

Relationships

Fitting subgroup ofC27×AGL(1,ℤ/9), 3−1+2×D27, #13091025
maximal subgroup ofC2×C27×3−1+2, C27×AGL(1,ℤ/9), 3−1+2×D27, …
maximal normal subgroup ofC2×C27×3−1+2, C27×AGL(1,ℤ/9), 3−1+2×D27, …
maximal quotient ofC2×C27×3−1+2, C27×C32⋊C9, C27×C9⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 286);