Finite Group Databasebeta

← Prev (#89076) Next (#89078) →

D₄×C₃⋊(C₃⋊D₃) #89077

order	432
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃³

Aliases

D₄×C₃⋊(C₃⋊D₃)

Splittings

Fitting	C₃³×D₄ ⋊ C₂
Sylow	C₃³ ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂×C₃³	C₂³
Fitting	C₃³×D₄	C₂
hypercenter	D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×D₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ D₄

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 13 points:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11723547, #11723553, #11723564, …
maximal normal subgroup of	#11723547, #11723553, #11723564, …
maximal quotient of	#11723919, C₂²⋊C₄×C₃⋊(C₃⋊D₃), #11723922, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 724);