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C₃×#6411 #10620594

order	672
isoclinism	isoclinic class of #1141
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Hall	C₃×C₇ ⋊ #104
Sylow	C₇ ⋊ C₃×#104

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₃×C₄
Frattini	C₂×C₄	C₂×C₃×D₇
Fitting, prime radical	C₃×C₇×#41	C₂
socle	C₂×C₃×C₇	C₂²⋊C₄
cosocle	C₂×C₄×C₇	C₂²×C₃
hypercenter	C₃×#41	D₇
hypocenter	C₇	C₃×#104
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×#41

88 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)(4,5,7,9,10,8,6)(11,13)(12,15)(14,17)(16,18);(5,6)(7,8)(9,10)(11,12,14,16,15,13,18,17)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#6411, #13054796, #13054798, …
maximal normal subgroup of	D₃×#6411, #13054796, #13054798, …
maximal quotient of	C₃×#89327, C₃×D₄⋊(C₇⋊C₈), C₃×Q₈⋊(C₇⋊C₈), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 176);