C₇×#4050 #11765148

order	1120
isoclinism	isoclinic class of #677
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4050

Splittings

Fitting	C₂×C₅×C₇×C₈ ⋊ C₂
Hall	C₅×C₇ ⋊ #135
Sylow	C₅ ⋊ C₇×#135

Subgroups

	subgroup	G/H
center	C₄×C₇	D₂₀
commutator, cosocle	C₄×C₅	C₂³×C₇
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₂×C₅×C₇×C₈	C₂
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	C₂×C₇×C₈	D₅
hypocenter	C₅	C₇×#135
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 560

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4,5,6,7)(8,9,11,12,10)(13,16,17,24,18,25,15,22)(14,20,19,23,21,27,26,28);(13,15,18,17)(14,19,21,26)(16,22,25,24)(20,23,27,28);(9,10)(11,12)(13,14)(15,19)(16,23)(17,26)(18,21)(20,24)(22,27)(25,28)

Conjugacy classes & character table

322 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 754);

C7×#4050 #11765148