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C₅×#1159 #10587969

order	560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₅×C₇×D₄ ⋊ C₂
Hall	C₅×C₇ ⋊ #41
Sylow	C₇ ⋊ C₅×#41

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂³×C₅
Frattini	C₂	C₂²×C₅×D₇
Fitting	C₅×C₇×D₄	C₂
socle	C₂×C₅×C₇	C₂³
hypercenter	C₅×D₄	D₇
hypocenter	C₇	C₅×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₅×D₄

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)(6,7,9,11,12,10,8)(14,18)(15,19);(13,15,16,19)(14,17,18,20);(7,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11764926, #11764932, #11764938, …
maximal normal subgroup of	#11764926, #11764932, #11764938, …
maximal quotient of	C₅×#6440, C₅×#6441, C₅×#6448, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(560, 141);