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C₇×#239 #66722

order	336
automorphism	Aut(G) = C₂²×C₃×D₃×S₄ (order 1728) · Inn(G) = C₂²×D₃ · Out(G) = C₂²×C₃×D₃
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₃×C₇×Q₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #41
Sylow	C₃ ⋊ C₇×#41

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂³×C₇
Frattini	C₂	C₂²×D₃×C₇
Fitting	C₃×C₇×Q₈	C₂
socle	C₂×C₃×C₇	C₂³
hypercenter	C₇×Q₈	D₃
hypocenter	C₃	C₇×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₇×Q₈

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

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

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

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10621066, #10621072, #10621079, …
maximal normal subgroup of	#10621066, #10621072, #10621079, …
maximal quotient of	C₇×#890, C₇×#892, C₇×#893, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 191);