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C₃²×C₇⋊C₈ #92620

order	504
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₃²×C₈ · Schur multiplier (H₂) = C₃

Aliases

C₃²×C₇⋊C₈

Splittings

derived, Sylow	C₇ ⋊ C₃²×C₈
Hall	C₃²×C₇ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₄	D₇
commutator, hypocenter	C₇	C₃²×C₈
Frattini	C₄	C₃²×D₇
Fitting	C₃²×C₄×C₇	C₂
socle, prime radical	C₂×C₃²×C₇	C₄
cosocle	C₄×C₇	C₂×C₃²
LCS	G ▷ C₇
UCS	C₁ ◁ C₃²×C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 504

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×C₇⋊C₈, C₃×D₂₁⋊C₈, C₃×#66558, …
maximal normal subgroup of	C₃×D₃×C₇⋊C₈, C₃×D₂₁⋊C₈, C₃×#66558, …
maximal quotient of	C₃²×C₇⋊C₁₆, C₂×C₃²×C₇⋊C₈, C₃×C₉×C₇⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 46);