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C₇×#688 #10587889

order	560
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₅×C₇×Q₈ ⋊ C₂
Hall	C₅×C₇ ⋊ SD₁₆
Sylow	C₅ ⋊ C₇×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂²×C₇
Frattini	C₄	C₂×C₇×D₅
Fitting	C₅×C₇×Q₈	C₂
socle	C₂×C₅×C₇	D₄
prime radical	C₇×D₂₀	C₂
hypercenter	C₇×Q₈	D₅
hypocenter	C₅	C₇×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×Q₈

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 280

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3,4,5,6,7)(8,9,11,12,10)(13,14,16,18)(15,20,19,17);(9,10)(11,12)(14,17)(15,19)(18,20)

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#688, #11764950, #11764953, …
maximal normal subgroup of	D₇×#688, #11764950, #11764953, …
maximal quotient of	C₇×#3940, C₇×#3941, C₇×#3967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(560, 61);