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C₂×3₋¹⁺²⋊Q₈ #88705

order	432
isoclinism	isoclinic class of 3₋¹⁺²⋊Q₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×3₋¹⁺²⋊Q₈
unambiguous semidirect product	(C₂×3₋¹⁺²)⋊Q₈
unambiguous semidirect product	(C₂×Dic₁₈)⋊C₃

Splittings

3₋¹⁺² ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center	C₂²	C₂×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₉	C₂³×C₃
Frattini	C₂×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×D₃
prime radical	C₂²×C₃²	C₂×D₃
hypercenter	C₂×C₄	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×Q₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄

122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#63368⋊C₃, #63376⋊C₃, #63382⋊C₃, …
maximal normal subgroup of	#63368⋊C₃, #63376⋊C₃, #63382⋊C₃, …
maximal quotient of	C₄×3₋¹⁺²⋊Q₈, #63431⋊C₃, #63432⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 352);