C₃×C₂₇² #14546550

order	2187
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 51018336
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃²×C₂₇

Aliases

abelian

C₃×C₂₇²

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉²	C₃³
socle, prime radical	C₃³	C₉²
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₉² ▷ C₃² ▷ C₁

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 2187

A minimal faithful permutation representation — 3 generators on 57 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);(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54);(55,56,57)

Conjugacy classes & character table

2187 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃×C₂₇², C₃×C₂₇×C₈₁, C₉×C₂₇², …
maximal normal subgroup of	C₂×C₃×C₂₇², C₃×C₂₇×C₈₁, C₉×C₂₇², …
maximal quotient of	C₂×C₃×C₂₇², C₃×C₂₇×C₈₁, C₉×C₂₇², …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 409);

C3×C272 #14546550