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C₇×#713 #10587981

order	560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₅×C₇×Q₈ ⋊ C₂
Hall	C₅×C₇ ⋊ #41
Sylow	C₅ ⋊ C₇×#41

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂³×C₇
Frattini	C₂	C₂²×C₇×D₅
Fitting	C₅×C₇×Q₈	C₂
socle	C₂×C₅×C₇	C₂³
hypercenter	C₇×Q₈	D₅
hypocenter	C₅	C₇×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₇×Q₈

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₇×#4010, #11764686, #11764952, …
maximal normal subgroup of	C₇×#4010, #11764686, #11764952, …
maximal quotient of	C₇×#4036, C₇×#4038, C₇×#4039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(560, 153);