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C₃×#4248 #92672

order	504
isoclinism	isoclinic class of #4245
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₃²×C₇ ⋊ Q₈
Sylow	C₃² ⋊ Dic₁₄
Sylow	C₇ ⋊ C₃×Dic₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×D₇
commutator, cosocle	C₂×C₃×C₇	C₂²×C₃
Frattini	C₂	C₃×D₃×D₇
Fitting, socle, prime radical	C₂×C₃²×C₇	C₂²
hypocenter	C₃×C₇	C₃×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₃

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 168

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

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

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#4248, #11759435, #11759436, …
maximal normal subgroup of	D₃×#4248, #11759435, #11759436, …
maximal quotient of	C₃×Dic₇⋊Dic₃, C₃×#66577, C₃×Dic₃⋊Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 98);