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Q₈⋊C₈×C₁₆₇ #24922727

order	10688
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	Q₈⋊C₈×C₁₆₇
unambiguous semidirect product	Q₈⋊(C₈×C₁₆₇)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₆₇	D₄
commutator	C₄	C₂×C₈×C₁₆₇
Frattini, cosocle	C₄²	C₂²×C₁₆₇
socle, prime radical	C₂²×C₁₆₇	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₆₇ ◁ C₄²×C₁₆₇ ◁ G

70 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

4676 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	Q₈⋊C₈×D₁₆₇, #36322519, #36322524, …
maximal subgroup of	C₂×Q₈⋊C₈×C₁₆₇, #1476×C₁₆₇, #1477×C₁₆₇, …
maximal normal subgroup of	C₂×Q₈⋊C₈×C₁₆₇, #1476×C₁₆₇, #1477×C₁₆₇, …
maximal quotient of	#1271×C₁₆₇, #1272×C₁₆₇, #1279×C₁₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10688, 7);