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C₃×#1161 #66712

order	336
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

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

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₇
Fitting	C₃×C₇×Q₈	C₂
socle	C₂×C₃×C₇	C₂³
hypercenter	C₃×Q₈	D₇
hypocenter	C₇	C₃×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×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:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10621064, #10621070, #10621076, …
maximal normal subgroup of	#10621064, #10621070, #10621076, …
maximal quotient of	C₃×#6453, C₃×#6455, C₃×#6456, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 181);