C₃×#11714262 #72227486

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#11714262

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#85559
commutator, cosocle, hypocenter	#2736⋊C₃	C₂×C₃
Frattini	C₂³	C₃×#1020
Fitting	C₃×#2736	D₃
socle	C₂³×C₃	#1020
prime radical	C₃×#2736⋊C₃	C₂
LCS	G ▷ #2736⋊C₃
UCS	C₁ ◁ C₂×C₃

383 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 20 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) = 35
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

60 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((1,2,3)(4,11,16,5,8,9,7,13)(6,17,14,10,15,12,19,18)(20,23,24,35)(21,28,27,34)(22,26,29,30)(25,32,33,31), (6,7,14)(9,10,17)(11,18,12)(15,16,19)(21,22,25)(23,31,33)(24,26,30)(27,32,34)(28,35,29));