C₅×#6452 #11765047

order	1120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#6452

Splittings

Hall	C₅×C₇ ⋊ #125
Sylow	C₇ ⋊ C₅×#125

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂³×C₅
Frattini	C₂²	C₂²×C₅×D₇
Fitting	C₅×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
hypercenter	C₅×C₄⋊C₄	D₇
hypocenter	C₇	C₅×#125
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

112 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 1120

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

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 653);