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C₂×(C₂²⋊C₉)⋊C₄ #63693

order	288
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = D₄×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 1728) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₃×D₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×(C₂²⋊C₉)⋊C₄
unambiguous semidirect product	(C₂×C₂²⋊C₉)⋊C₄

Splittings

C₂²⋊C₉ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	(C₂²⋊C₉)⋊C₂
commutator, hypocenter	C₂²⋊C₉	C₂×C₄
Frattini	C₂×C₃	C₂×S₄
Fitting, socle, prime radical	C₂⁴×C₃	D₃
cosocle	C₂×C₂²⋊C₉	C₂²
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₂²

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×(C₂²⋊C₉)⋊C₄, ((C₂²⋊C₉)⋊C₄)⋊C₄, C₄⋊((C₂²⋊C₉)⋊C₄), …
maximal normal subgroup of	C₄×(C₂²⋊C₉)⋊C₄, ((C₂²⋊C₉)⋊C₄)⋊C₄, C₄⋊((C₂²⋊C₉)⋊C₄), …
maximal quotient of	C₂×(C₂²⋊C₉)⋊C₈, #10589511, C₄×(C₂²⋊C₉)⋊C₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 341);