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C3×3−1+2⋊C9 #10624696

order729
nilpotency class3
isoclinismisoclinic class of AΣL(1,𝔽27)
automorphism?
homologyAbelianization (H1) = C32×C9 · Schur multiplier (H2) = C33

Aliases

direct productC3×3−1+2⋊C9

Subgroups

subgroupG/H
center, socleC33H(3,𝔽3)
commutatorC32C32×C9
Frattini, cosocleC33C33
prime radicalC34C32
LCSG ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C33 ◁ C34 ◁ G
LECSG ▷ C33 ▷ C3 ▷ C1

286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 243

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊((3−1+2⋊C9)⋊C2)
Fitting subgroup ofD3×3−1+2⋊C9, C3×(3−1+2⋊C9)⋊C2, C3⋊((3−1+2⋊C9)⋊C2)
maximal subgroup ofC2×C3×3−1+2⋊C9, D3×3−1+2⋊C9, C3×(3−1+2⋊C9)⋊C2, …
maximal normal subgroup ofC2×C3×3−1+2⋊C9, D3×3−1+2⋊C9, C3×(3−1+2⋊C9)⋊C2, …
maximal quotient ofC2×C3×3−1+2⋊C9, C3×3−1+2⋊C27, #14546596, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 256);