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Q₃₂×D₁₆₇ #24923695

order	10688
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Q₃₂×D₁₆₇

Splittings

Fitting	Q₃₂×C₁₆₇ ⋊ C₂
Sylow	C₁₆₇ ⋊ C₂×Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₈×D₁₆₇
commutator, cosocle	C₈×C₁₆₇	C₂³
Frattini	C₈	C₂²×D₁₆₇
Fitting	Q₃₂×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂×D₈
prime radical	C₂×D₁₆₇	D₈
hypercenter	Q₃₂	D₁₆₇
hypocenter	C₁₆₇	C₂×Q₃₂
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

82 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

935 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₆₄×D₁₆₇, Q₆₄×D₁₆₇, #36326454, …
maximal normal subgroup of	SD₆₄×D₁₆₇, Q₆₄×D₁₆₇, #36326454, …
maximal quotient of	#358×D₁₆₇, #366×D₁₆₇, #36325994, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10688, 975);