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Q₃₂×D₈₉ #19152024

order	5696
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Q₃₂×D₈₉

Splittings

Fitting	Q₃₂×C₈₉ ⋊ C₂
Sylow	C₈₉ ⋊ C₂×Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₈×D₈₉
commutator	C₈×C₈₉	C₂³
Frattini	C₈	C₂²×D₈₉
Fitting	Q₃₂×C₈₉	C₂
socle	C₂×C₈₉	C₂×D₈
hypercenter	Q₃₂	D₈₉
hypocenter	C₈₉	C₂×Q₃₂
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

82 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

506 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₆₄×D₈₉, Q₆₄×D₈₉, #26022568, …
maximal normal subgroup of	SD₆₄×D₈₉, Q₆₄×D₈₉, #26022568, …
maximal quotient of	#358×D₈₉, #366×D₈₉, #26022108, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5696, 975);