C₅×#5237 #11750736

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#5237

Splittings

Hall	C₃×C₅ ⋊ #413
Sylow	C₃ ⋊ C₅×#413

Subgroups

	subgroup	G/H
center	C₄×C₅	#217
commutator	C₂²×C₃	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#41	C₂
socle	C₂×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#41	D₃
hypocenter	C₃	C₅×#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#41

300 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#67700, C₅×#67693, #14279104, …
maximal normal subgroup of	C₅×#67700, C₅×#67693, #14279104, …
maximal quotient of	C₅×#71571, C₅×#71600, C₅×#71572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4931);