Finite Group Databasebeta

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C₇×#215 #66618

order	336
isoclinism	isoclinic class of #213
automorphism	Aut(G) = C₂²×C₃×D₃×D₄ (order 576) · Inn(G) = #67 · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

Splittings

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

Subgroups

	subgroup	G/H
center	C₂×C₇	#67
commutator, cosocle	C₃×C₄	C₂²×C₇
Frattini	C₄	C₂×D₃×C₇
Fitting	C₃×C₇×Q₈	C₂
socle	C₂×C₃×C₇	D₄
prime radical	C₇×D₁₂	C₂
hypercenter	C₇×Q₈	D₃
hypocenter	C₃	C₇×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#215, #10621064, #10621067, …
maximal normal subgroup of	D₇×#215, #10621064, #10621067, …
maximal quotient of	C₇×#808, C₇×#809, C₇×#835, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 87);