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C₂²×C₂²⋊C₉ #3767

order	144
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×D₃×S₄ (order 432) · Inn(G) = A₄ · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂²

Aliases

C₂²×C₂²⋊C₉

Splittings

derived	C₂² ⋊ C₂²×C₉
Sylow	C₂⁴ ⋊ C₉

Subgroups

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

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of	A₄×(C₂²⋊C₉)⋊C₂, ((C₂²⋊C₉)⋊C₂)⋊A₄, #11920004, …
maximal subgroup of	C₂×(C₂²⋊C₉)⋊C₄, #63694, C₂×C₄×C₂²⋊C₉, …
maximal normal subgroup of	C₂×(C₂²⋊C₉)⋊C₄, #63694, C₂×C₄×C₂²⋊C₉, …
maximal quotient of	C₂×C₄×C₂²⋊C₉, D₄×C₂²⋊C₉, C₂²×Q₈⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 110);