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C₂²×C₃×C₂²⋊C₉ #88901

order	432
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₉ · Schur multiplier (H₂) = C₂²×C₃

Aliases

C₂²×C₃×C₂²⋊C₉

Splittings

derived	C₂² ⋊ C₂²×C₃×C₉
Sylow	C₂⁴ ⋊ C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₃²	A₄
commutator, hypocenter	C₂²	C₂²×C₃×C₉
Frattini	C₃	C₂²×C₃×A₄
Fitting, socle, prime radical	C₂⁴×C₃²	C₃
cosocle	C₂²×C₃	C₂²×C₃²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂²×C₃²

214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72245336, C₂×#72220173
maximal subgroup of	C₂×C₃×(C₂²⋊C₉)⋊C₄, C₃×#63694, C₂×C₃⋊((C₂²⋊C₉)⋊C₄), …
maximal normal subgroup of	C₂×C₃×(C₂²⋊C₉)⋊C₄, C₃×#63694, C₂×C₃⋊((C₂²⋊C₉)⋊C₄), …
maximal quotient of	C₂×C₃×C₄×C₂²⋊C₉, C₃×D₄×C₂²⋊C₉, C₂²×C₃×Q₈⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 548);