C₅×#5570 #11755955

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

Aliases

direct product

C₅×#5570

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 230 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 230 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);(17,20,21,18)(19,22,23,24);(17,19,21,23)(18,22,20,24);(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,22,23,24)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71030, C₅×#68252, #14289201, …
maximal normal subgroup of	C₅×#71030, C₅×#68252, #14289201, …
maximal quotient of	C₅×Q₈×C₄⋊Dic₃, C₅×#82461, C₅×#73640, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10150);