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C₇×#837 #10620635

order	672
isoclinism	isoclinic class of #213
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Hall	C₃×C₇ ⋊ #104
Sylow	C₃ ⋊ C₇×#104

Subgroups

	subgroup	G/H
center	C₄×C₇	#67
commutator	C₃×C₄	C₂×C₄×C₇
Frattini	C₂×C₄	C₂×D₃×C₇
Fitting, prime radical	C₃×C₇×#41	C₂
socle	C₂×C₃×C₇	C₂²⋊C₄
cosocle	C₂×C₃×C₄	C₂²×C₇
hypercenter	C₇×#41	D₃
hypocenter	C₃	C₇×#104
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#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:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#837, #13054796, #13054797, …
maximal normal subgroup of	D₇×#837, #13054796, #13054797, …
maximal quotient of	C₇×#4532, C₇×D₄⋊(C₃⋊C₈), C₇×#4535, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 217);