C₃×#11714256 #72227491

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

solvable

Aliases

direct product

C₃×#11714256

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#85559
commutator, cosocle, hypocenter	#73292	C₂×C₃
Frattini	C₂³	C₃×#1020
Fitting	C₃×#2840	D₃
socle	C₂³×C₃	#1020
prime radical	C₃×#73292	C₂
LCS	G ▷ #73292
UCS	C₁ ◁ C₂×C₃

387 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 22 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)(4,15,16,12,8,9,13,10)(5,6,19,7,11,14,18,17)(20,27,33,22,31,26,35,30)(21,28,29,23,34,32,25,24);(6,7,16)(9,10,19)(12,18,15)(13,14,17)(21,25,32)(22,28,27)(23,24,31)(26,29,33)(30,35,34)

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227492

GAP

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

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