Finite Group Databasebeta

← Prev (#66684) Next (#66686) →

C₂×D₃×Dic₇ #66685

order	336
isoclinism	isoclinic class of D₃×D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₃×Dic₇

Splittings

derived, Hall	C₃×C₇ ⋊ C₂²×C₄
Sylow	C₇ ⋊ C₂×C₄×D₃
Sylow	C₃ ⋊ C₂²×Dic₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂²×C₄
Frattini	C₂	C₂×D₃×D₇
Fitting, socle	C₂²×C₃×C₇	C₂²
prime radical	C₂²×D₃×C₇	C₂
cosocle	C₂×C₃×C₇	C₂³
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂²

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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×D₃×Dic₇, #10620960, D₃×Dic₇⋊C₄, …
maximal normal subgroup of	C₄×D₃×Dic₇, #10620960, D₃×Dic₇⋊C₄, …
maximal quotient of	C₂×D₃×C₇⋊C₈, #10620848, D₃×#1136, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 154);