C₃×C₃⋊#92338 #72243436

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₃×C₃⋊#92338

Splittings

Fitting, Sylow	C₃²×C₃²⋊H(3,𝔽₃) ⋊ C₂
derived	C₃×C₃²⋊H(3,𝔽₃) ⋊ C₂×C₃

Subgroups

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

20299 conjugacy classes of subgroups; only the 542 normal subgroups are shown.

normal subgroups only

This group has 20299 conjugacy classes of subgroups — too many to draw. Only its 542 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 — 5 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72243459, C₃×#72214065, C₃×#72224516
maximal normal subgroup of	C₃×#72214065, C₃×#72224516

GAP

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

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