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C₃×#4247 #92671

order	504
isoclinism	isoclinic class of #4245
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Hall	C₃²×C₇ ⋊ D₄
Sylow	C₃² ⋊ C₂²⋊D₇
Sylow	C₇ ⋊ C₃×D₁₂

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×D₇
commutator, cosocle	C₂×C₃×C₇	C₂²×C₃
Frattini	C₂	C₃×D₃×D₇
Fitting, socle	C₂×C₃²×C₇	C₂²
hypocenter	C₃×C₇	C₃×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₃

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#4247, #11759430, #11759431, …
maximal normal subgroup of	D₃×#4247, #11759430, #11759431, …
maximal quotient of	C₃×#66562, C₃×#66567, C₃×#66568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 97);