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C₂³×C₇⋊C₉ #92634

order	504
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

C₂³×C₇⋊C₉

Splittings

derived, Sylow	C₇ ⋊ C₂³×C₉
Hall	C₂³×C₇ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₃	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₂³×C₉
Frattini	C₃	C₂³×ΓL(1,𝔽₈)
Fitting, socle, prime radical	C₂³×C₃×C₇	C₃
cosocle	C₃×C₇	C₂³×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₃

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 504

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂²⋊Dic₇)⋊C₉, C₂²⋊C₄×C₇⋊C₉, C₂²×Dic₇⋊C₉, …
maximal normal subgroup of	(C₂²⋊Dic₇)⋊C₉, C₂²⋊C₄×C₇⋊C₉, C₂²×Dic₇⋊C₉, …
maximal quotient of	C₂²×C₄×C₇⋊C₉, C₂×D₄×C₇⋊C₉, C₂×Q₈×C₇⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 60);