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C₃×D₃×Dic₇ #92667

order	504
isoclinism	isoclinic class of D₃×D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₃×D₃×Dic₇

Splittings

derived	C₃×C₇ ⋊ C₂×C₃×C₄
Hall	C₃²×C₇ ⋊ C₂×C₄
Sylow	C₃² ⋊ C₂×Dic₇
Sylow	C₇ ⋊ C₃×C₄×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂×C₃×C₄
Frattini	C₂	C₃×D₃×D₇
Fitting, socle	C₂×C₃²×C₇	C₂²
prime radical	C₂×C₃×D₃×C₇	C₂
cosocle	C₂×C₃×C₇	C₂²×C₃
LCS	G ▷ 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)(4,5,6)(7,10,9,8)(12,13)(14,15)(16,17);(5,6)(11,12,14,16,17,15,13)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃²×Dic₇, D₃×D₂₁⋊C₄, D₃×#4247, …
maximal normal subgroup of	D₃²×Dic₇, D₃×D₂₁⋊C₄, D₃×#4247, …
maximal quotient of	C₃×D₃×C₇⋊C₈, C₃×#66558, C₃×Dic₃×Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 93);