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C₂×C₂²⋊C₉ #619

order	72
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×S₄ (order 72) · Inn(G) = A₄ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = C₂×#72213207
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₂²⋊C₉
unambiguous semidirect product	C₂³⋊C₉

Splittings

derived	C₂² ⋊ C₂×C₉
Sylow	C₂³ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	A₄
commutator, hypocenter	C₂²	C₂×C₉
Frattini	C₃	C₂×A₄
Fitting, socle, prime radical	C₂³×C₃	C₃
cosocle	C₂²×C₃	C₂×C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₃

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

derived subgroup of	(C₂²⋊C₉)⋊Q₈, C₄⋊((C₂²⋊C₉)⋊C₂), #63694, …
maximal subgroup of	(C₂²⋊C₉)⋊C₄, C₄×C₂²⋊C₉, C₂×(C₂²⋊C₉)⋊C₂, …
maximal normal subgroup of	(C₂²⋊C₉)⋊C₄, C₄×C₂²⋊C₉, C₂×(C₂²⋊C₉)⋊C₂, …
maximal quotient of	C₄×C₂²⋊C₉, C₂×Q₈⋊C₉, #41⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 16);