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D₃×#634 #89013

order	432
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Splittings

C₃³ ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃³	C₂³
Fitting	C₃³×C₄	C₂²
prime radical	C₂×C₃²×D₃	C₂²
hypercenter	C₄	D₃×C₃⋊D₃
hypocenter	C₃³	C₂×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3716, D₃×#3719, #11722886, …
maximal normal subgroup of	D₃×#3716, D₃×#3719, #11722886, …
maximal quotient of	#11723389, #11723394, #11723395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 660);