C₃×#11711413 #72227490

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₄

solvable

Aliases

direct product

C₃×#11711413

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	#1020⋊C₂
commutator, cosocle, hypocenter	#543⋊C₃	C₂²×C₃
Frattini	C₂⁴	C₂×C₃×S₄
Fitting	C₃×#2029	D₃
prime radical	C₃×#2029⋊C₃	C₂
LCS	G ▷ #543⋊C₃
UCS	C₁ ◁ C₂²×C₃

490 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 96

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

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

Conjugacy classes & character table

81 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)(4,13,7,6)(5,18,11,12)(8,14,17,15)(9,19,16,10)(20,28,23,34,24,35,33,25)(21,32,26,22,27,29,31,30), (4,5)(6,16,13,12,7,18)(8,9)(10,17,19,15,11,14)(22,23,32)(25,26,35)(28,34,31)(29,30,33));