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C₂³×C₂²⋊C₉ #64189

order	288
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂³×C₂²⋊C₉

Splittings

derived	C₂² ⋊ C₂³×C₉
Sylow	C₂⁵ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₃	A₄
commutator, hypocenter	C₂²	C₂³×C₉
Frattini	C₃	C₂³×A₄
Fitting, socle, prime radical	C₂⁵×C₃	C₃
cosocle	C₂²×C₃	C₂³×C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂³×C₃

324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

derived subgroup of	A₄×#63694, #72269060, A₄×#63694⋊C₃
maximal subgroup of	#10589534, C₂²⋊C₄×C₂²⋊C₉, C₂²×(C₂²⋊C₉)⋊C₄, …
maximal normal subgroup of	#10589534, C₂²⋊C₄×C₂²⋊C₉, C₂²×(C₂²⋊C₉)⋊C₄, …
maximal quotient of	C₂²×C₄×C₂²⋊C₉, C₂×D₄×C₂²⋊C₉, C₂³×Q₈⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 837);