C₅×#4897 #11750396

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

Aliases

direct product

C₅×#4897

Splittings

Fitting	C₃×C₅×#108 ⋊ C₂
Hall	C₃×C₅ ⋊ #472
Sylow	C₃ ⋊ C₅×#472

Subgroups

	subgroup	G/H
center	C₂×C₅	#895
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#108	C₂
socle	C₂×C₃×C₅	#121
hypercenter	C₅×#108	D₃
hypocenter	C₃	C₅×#472
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#108

216 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 74 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)(9,12,14,18,15,17,13,10)(11,19,21,24,22,16,20,23);(9,11,14,21,15,22,13,20)(10,16,17,24,18,19,12,23);(7,8)(9,10)(11,16)(12,13)(14,17)(15,18)(19,22)(20,23)(21,24)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#69242, C₅×#69258, #14316423, …
maximal normal subgroup of	C₅×#69242, C₅×#69258, #14316423, …
maximal quotient of	C₅×#68576, C₅×#68586, C₅×#68579, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4591);