C₅×#5179 #11750678

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

Aliases

direct product

C₅×#5179

Splittings

Fitting	C₂×C₃×C₅×Q₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #510
Sylow	C₃ ⋊ C₅×#510

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#213
commutator, cosocle	C₃×C₈	C₂³×C₅
Frattini	C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×Q₁₆	C₂
socle	C₂×C₃×C₅	C₂×D₈
prime radical	C₅×#904	C₂
hypercenter	C₂×C₅×Q₁₆	D₃
hypocenter	C₃	C₅×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×Q₁₆

164 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 40 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#70917, C₅×#69264, #14315791, …
maximal normal subgroup of	C₅×#70917, C₅×#69264, #14315791, …
maximal quotient of	C₅×#70598, C₅×#72061, C₅×#70596, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4873);