C₅×#6481 #11765076

order	1120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#6481

Splittings

Fitting	C₅×C₇×Q₁₆ ⋊ C₂
Hall	C₅×C₇ ⋊ #137
Sylow	C₇ ⋊ C₅×#137

Subgroups

	subgroup	G/H
center	C₂×C₅	D₄×D₇
commutator, cosocle	C₄×C₇	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₇
Fitting	C₅×C₇×Q₁₆	C₂
socle	C₂×C₅×C₇	C₂×D₄
prime radical	C₅×#1161	C₂
hypercenter	C₅×Q₁₆	D₇
hypocenter	C₇	C₅×#137
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×Q₁₆

120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 560

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

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 682);