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C₄×D₃×D₇ #66678

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

Aliases

C₄×D₃×D₇

Splittings

Fitting	C₃×C₄×C₇ ⋊ C₂²
derived, Hall	C₃×C₇ ⋊ C₂²×C₄
Sylow	C₇ ⋊ C₂×C₄×D₃
Sylow	C₃ ⋊ C₂×C₄×D₇

Subgroups

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

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₈×D₇, D₇×ΓL(1,𝔽₂₅), D₃×#1131, …
maximal normal subgroup of	D₃×C₈×D₇, D₇×ΓL(1,𝔽₂₅), D₃×#1131, …
maximal quotient of	D₃×C₈×D₇, D₇×ΓL(1,𝔽₂₅), D₃×#1131, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 147);