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C₉×C₂₇ #6863

order	243
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#72277433 (order 8748) · Inn(G) = C₁ · Out(G) = C₂×#72277433
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₉

Aliases

C₉×C₂₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₉	C₃²
socle, prime radical	C₃²	C₃×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 2 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);(28,29,30,31,32,33,34,35,36)

Conjugacy classes & character table

243 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₉×C₂₇, #14546221, #14546222, …
derived subgroup of	C₉⋊D₂₇, C₉⋊Dic₂₇, D₉×D₂₇, …
Frattini subgroup of	C₂₇×C₈₁, #14546221, #14546222, …
Fitting subgroup of	C₉×D₂₇, D₉×C₂₇, C₉⋊D₂₇, …
maximal subgroup of	C₉×D₂₇, D₉×C₂₇, C₉⋊D₂₇, …
maximal normal subgroup of	C₉×D₂₇, D₉×C₂₇, C₉⋊D₂₇, …
maximal quotient of	C₂×C₉×C₂₇, C₂₇², #10624443, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(243, 10);