C₅×#5753 #11756138

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#5753

Splittings

Fitting	C₃×C₅×#131 ⋊ C₂
Hall	C₃×C₅ ⋊ #578
Sylow	C₃ ⋊ C₅×#578

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#131	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₅×#131	D₃
hypocenter	C₃	C₅×#578
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₅×#41 ◁ C₅×#131

496 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 214 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,4,5)(6,7,8)(9,13,12,24,15,27,14,26)(10,18,28,38,19,33,17,25)(11,21,23,37,22,36,20,34)(16,30,29,39,32,40,31,35);(9,12,15,14)(10,17,19,28)(11,20,22,23)(13,24,27,26)(16,29,32,31)(18,25,33,38)(21,34,36,37)(30,39,40,35);(9,11,15,22)(10,16,19,32)(12,23,14,20)(13,21,27,36)(17,31,28,29)(18,30,33,40)(24,37,26,34)(25,35,38,39);(7,8)(9,10)(11,16)(12,17)(13,25)(14,28)(15,19)(18,26)(20,29)(21,35)(22,32)(23,31)(24,33)(27,38)(30,34)(36,39)(37,40)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71095, C₅×#69127, #14316474, …
maximal normal subgroup of	C₅×#71095, C₅×#69127, #14316474, …
maximal quotient of	C₅×#83806, C₅×#83785, C₅×#78571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10333);