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C₇×#704 #10587928

order	560
isoclinism	isoclinic class of #700
automorphism	Aut(G) = C₂×C₃×D₄×AGL(1,𝔽₅) (order 960) · Inn(G) = C₂×AGL(1,𝔽₅) · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₅×C₇ ⋊ M₁₆
Sylow	C₅ ⋊ C₇×M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄×C₇
Frattini	C₂	C₂×C₇×AGL(1,𝔽₅)
Fitting, prime radical	C₂²×C₅×C₇	C₄
socle	C₂×C₅×C₇	C₂×C₄
cosocle	Dic₅	C₂²×C₇
hypercenter	C₂²×C₇	AGL(1,𝔽₅)
hypocenter	C₅	C₇×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11764637, #11764638, C₇×#4005, …
maximal normal subgroup of	#11764637, #11764638, C₇×#4005, …
maximal quotient of	C₇×#4002, C₇×#4004, C₇×C₂²⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(560, 100);