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Q₈×C₇⋊C₉ #92595

order	504
isoclinism	isoclinic class of D₄×ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₁

Aliases

Q₈×C₇⋊C₉

Splittings

Hall	C₇×Q₈ ⋊ C₉
Sylow	C₇ ⋊ Q₈×C₉

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₃	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂²×C₉
Fitting	C₃×C₇×Q₈	C₃
socle, prime radical, cosocle	C₂×C₃×C₇	C₂²×C₃
hypercenter	C₃×Q₈	ΓL(1,𝔽₈)
hypocenter	C₇	Q₈×C₉
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 504

A minimal faithful permutation representation — 2 generators on 24 points:

(2,3,4)(5,7,6)(8,10,11,15)(9,12,13,14)(16,20,24,18,23,19,21,17,22);(1,2,3,5,4,6,7)(8,9,11,13)(10,14,15,12)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1143⋊C₉, #1144⋊C₉, SD₁₆×C₇⋊C₉, …
maximal normal subgroup of	#1143⋊C₉, #1144⋊C₉, SD₁₆×C₇⋊C₉, …
maximal quotient of	C₄⋊C₄×C₇⋊C₉, C₂×Q₈×C₇⋊C₉, Q₈×C₇⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 21);