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C₃×#1143 #66602

order	336
isoclinism	isoclinic class of #1141
automorphism	Aut(G) = C₂²×D₄×AGL(1,𝔽₇) (order 1344) · Inn(G) = C₂²⋊D₇ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₃×C₇×Q₈ ⋊ C₂
Hall	C₃×C₇ ⋊ SD₁₆
Sylow	C₇ ⋊ C₃×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂²×C₃
Frattini	C₄	C₂×C₃×D₇
Fitting	C₃×C₇×Q₈	C₂
socle	C₂×C₃×C₇	D₄
prime radical	C₃×D₂₈	C₂
hypercenter	C₃×Q₈	D₇
hypocenter	C₇	C₃×SD₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1143, #10621066, #10621067, …
maximal normal subgroup of	D₃×#1143, #10621066, #10621067, …
maximal quotient of	C₃×#6382, C₃×#6383, C₃×#6409, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 71);