C₃×#14515301 #72227481

order	5760
isoclinism	isoclinic class of #14515301
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₄²

non-solvable

Aliases

direct product

C₃×#14515301

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂×C₃	#11757162
commutator, hypocenter	#14515301	C₃
Frattini	C₂	C₃×#11757162
Fitting	C₂⁵×C₃	A₅
cosocle	C₂⁵	C₃×A₅
LCS	G ▷ #14515301
UCS	C₁ ◁ C₂×C₃

278 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 27 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227488

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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