C₃×#14515304 #72227482

order	5760
isoclinism	isoclinic class of #14515304
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

non-solvable

Aliases

direct product

C₃×#14515304

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#11757163
commutator, hypocenter	#14515304	C₃
Frattini	C₂	C₃×#11757163
Fitting, socle	C₂⁵×C₃	A₅
cosocle	C₂⁵	C₃×A₅
LCS	G ▷ #14515304
UCS	C₁ ◁ C₂×C₃

278 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 10 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) = 37
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 37 points:

(1,9,8,5,10)(2,7,3,6,4)(11,14,24,17,30,12,16,18,23,31)(13,15)(19,32,20,21,33,25,34,26,27,29)(22,28);(1,2,4,3,5,8)(7,10)(11,14,22)(12,16,28)(13,19,32)(15,25,34)(17,20,33)(18,31,21)(23,26,29)(24,30,27)(35,37,36)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227488, #72227492, #72227496

GAP

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

gap> G := Group((1,9,8,5,10)(2,7,3,6,4)(11,14,24,17,30,12,16,18,23,31)(13,15)(19,32,20,21,33,25,34,26,27,29)(22,28), (1,2,4,3,5,8)(7,10)(11,14,22)(12,16,28)(13,19,32)(15,25,34)(17,20,33)(18,31,21)(23,26,29)(24,30,27)(35,37,36));