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C₂×C₃×C₂²⋊D₇ #66714

order	336
isoclinism	isoclinic class of Dic₁₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₃×C₂²⋊D₇

Splittings

Fitting	C₂³×C₃×C₇ ⋊ C₂
Hall	C₃×C₇ ⋊ C₂×D₄
Sylow	C₇ ⋊ C₂×C₃×D₄

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×D₇
commutator, cosocle	C₂×C₇	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₇
Fitting	C₂³×C₃×C₇	C₂
socle	C₂²×C₃×C₇	C₂²
hypercenter	C₂³×C₃	D₇
hypocenter	C₇	C₂×C₃×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620530, C₃×#6380, #10621095, …
maximal normal subgroup of	#10620530, C₃×#6380, #10621095, …
maximal quotient of	C₂×C₃×Dic₇⋊C₄, C₃×#6487, C₂×C₃×(C₂×D₇)⋊C₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 183);