C₃×#65207 #11749815

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#65207

Splittings

Fitting	C₃×C₅×Q₃₂ ⋊ C₂
Hall	C₃×C₅ ⋊ #510
Sylow	C₅ ⋊ C₃×#510

Subgroups

	subgroup	G/H
center	C₂×C₃	D₅×D₈
commutator, cosocle	C₅×C₈	C₂³×C₃
Frattini	C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×Q₃₂	C₂
socle	C₂×C₃×C₅	C₂×D₈
prime radical	C₃×#4065	C₂
hypercenter	C₃×Q₃₂	D₅
hypocenter	C₅	C₃×#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×Q₃₂

164 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 62 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)(4,5,7,8,6)(9,11,14,19)(10,15,18,30)(12,24,27,40)(13,29,25,22)(16,34,35,39)(17,36,28,32)(20,38,37,23)(21,33,31,26);(9,10,14,18)(11,20,19,37)(12,23,27,38)(13,28,25,17)(15,31,30,21)(16,33,35,26)(22,39,29,34)(24,36,40,32);(5,6)(7,8)(11,19)(15,30)(20,37)(21,31)(22,29)(24,40)(32,36)(34,39)

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10604139, #14315764, C₃×#10604138, …
maximal normal subgroup of	C₃×#10604139, #14315764, C₃×#10604138, …
maximal quotient of	C₃×#10600014, C₃×#10600015, C₃×#10600119, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4010);