Finite Group Databasebeta

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C92×3−1+2 #14552009

order2187
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C32×C92 · Schur multiplier (H2) = C34×C9

Aliases

direct productC92×3−1+2

Subgroups

subgroupG/H
centerC3×C92C32
commutatorC3C32×C92
Frattini, socle, cosocleC33C34
prime radicalC34C33
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C3×C92 ◁ G
LECSG ▷ C33 ▷ C1

1664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 729

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of3−1+2×C9⋊D9, C92×AGL(1,ℤ/9), #72243214, …
maximal subgroup of3−1+2×#6900, 3−1+2×C9⋊D9, C92×AGL(1,ℤ/9), …
maximal normal subgroup of3−1+2×#6900, 3−1+2×C9⋊D9, C92×AGL(1,ℤ/9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 5868);