H(3,𝔽₃)×C₃⋊AGL(1,ℤ/9) #72243662

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

H(3,𝔽₃)×C₃⋊AGL(1,ℤ/9)

Splittings

Fitting, Sylow

C₃×H(3,𝔽₃)×3₋¹⁺² ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃²×C₃⋊AGL(1,ℤ/9)
commutator, cosocle	C₃²×C₉	C₂×C₃³
Frattini	C₃²	C₃³×C₃⋊D₃
Fitting	C₃×H(3,𝔽₃)×3₋¹⁺²	C₂
socle	C₃³	C₃³×D₃
hypercenter	H(3,𝔽₃)	C₃⋊AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₂×C₃×H(3,𝔽₃)
LCS	G ▷ C₃²×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₃ ◁ H(3,𝔽₃)

4843 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 4843 conjugacy classes of subgroups — too many to draw. Only its 252 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) = 21
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243669, (C₃⋊D₃)⋊C₃×C₃⋊AGL(1,ℤ/9), D₃×H(3,𝔽₃)×AGL(1,ℤ/9), …
maximal normal subgroup of	(C₃⋊D₃)⋊C₃×C₃⋊AGL(1,ℤ/9), D₃×H(3,𝔽₃)×AGL(1,ℤ/9), #72254081
maximal quotient of	#72243669

GAP

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

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