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D₄⋊C₈×C₁₃₇ #22576860

order	8768
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	D₄⋊C₈×C₁₃₇
unambiguous semidirect product	D₄⋊(C₈×C₁₃₇)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃₇	D₄
commutator	C₄	C₂×C₈×C₁₃₇
Frattini, cosocle	C₄²	C₂²×C₁₃₇
socle	C₂²×C₁₃₇	C₂²⋊C₄
prime radical	C₂×D₄×C₁₃₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

3836 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	D₄⋊C₈×D₁₃₇, #31779168, #31779173, …
maximal subgroup of	C₂×D₄⋊C₈×C₁₃₇, #1476×C₁₃₇, #1477×C₁₃₇, …
maximal normal subgroup of	C₂×D₄⋊C₈×C₁₃₇, #1476×C₁₃₇, #1477×C₁₃₇, …
maximal quotient of	#1270×C₁₃₇, #1271×C₁₃₇, #1279×C₁₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8768, 6);