C₃²×(C₃⋊D₃)⋊H(3,𝔽₃) #72243494

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁸

Aliases

direct product

C₃²×(C₃⋊D₃)⋊H(3,𝔽₃)

Splittings

Fitting, Sylow

C₃²×C₃²⋊H(3,𝔽₃) ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃³	C₃×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃³	C₂×C₃⁴
Frattini	C₃²	C₃⁴×D₃
Fitting	C₃²×C₃²⋊H(3,𝔽₃)	C₂
socle	C₃⁴	C₃²×D₃
hypercenter	C₃⁴	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×C₃²×H(3,𝔽₃)
LCS	G ▷ C₃³ ▷ C₃²
UCS	C₁ ◁ C₃³ ◁ C₃⁴

18790 conjugacy classes of subgroups; only the 752 normal subgroups are shown.

normal subgroups only

This group has 18790 conjugacy classes of subgroups — too many to draw. Only its 752 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) = 24
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

378 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#72243516, C₃²×#11757702, C₃×#72224516
maximal normal subgroup of	#72243516, C₃²×#11757702, C₃×#72224516

GAP

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

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