C₅×#5569 #11755954

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#5569

Splittings

Hall	C₃×C₅ ⋊ #557
Sylow	C₃ ⋊ C₅×#557

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₃×C₅	C₂⁴
hypercenter	C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈

400 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 24
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#80580, C₅×#79700, #14472120, …
maximal normal subgroup of	C₅×#80580, C₅×#79700, #14472120, …
maximal quotient of	C₅×Q₈×Dic₃⋊C₄, C₅×#82461, C₅×#82460, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10149);