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C₂²×C₃×Dic₇ #66713

order	336
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×C₃×Dic₇

Splittings

derived, Sylow	C₇ ⋊ C₂²×C₃×C₄
Hall	C₃×C₇ ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₃	D₇
commutator, hypocenter	C₇	C₂²×C₃×C₄
Frattini	C₂	C₂²×C₃×D₇
Fitting, socle, prime radical	C₂³×C₃×C₇	C₂
cosocle	C₂×C₇	C₂³×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₃

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620529, C₃×C₇⋊#95, C₂×Dic₃×Dic₇, …
maximal normal subgroup of	#10620529, C₃×C₇⋊#95, C₂×Dic₃×Dic₇, …
maximal quotient of	C₂²×C₃×C₇⋊C₈, C₂×C₃×#1136, C₂×C₃×C₄×Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 182);