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C₃×(C₂²⋊C₉)⋊C₄ #88603

order	432
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = C₂²×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 864) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₃×(C₂²⋊C₉)⋊C₄

Splittings

C₂²⋊C₉ ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₂²⋊C₉)⋊C₂
commutator, hypocenter	C₂²⋊C₉	C₃×C₄
Frattini	C₂×C₃	C₃×S₄
Fitting, socle, prime radical	C₂³×C₃²	D₃
cosocle	C₂×C₂²⋊C₉	C₂×C₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₂×C₃

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 36

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

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11722261, (C₃⋊((C₂²⋊C₉)⋊C₂))⋊C₄, D₃×(C₂²⋊C₉)⋊C₄, …
maximal normal subgroup of	#11722261, (C₃⋊((C₂²⋊C₉)⋊C₂))⋊C₄, D₃×(C₂²⋊C₉)⋊C₄, …
maximal quotient of	C₃×(C₂²⋊C₉)⋊C₈, C₃×#63421, C₃×#63422, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 250);