C₃×#65480 #11750088

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#65480

Splittings

Fitting	C₂×C₃×C₅×Q₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #495
Sylow	C₅ ⋊ C₃×#495

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×Q₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×D₄₀	C₂
hypercenter	C₂×C₃×Q₁₆	D₅
hypocenter	C₅	C₃×#495
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₁₆

260 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 28
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3)(4,5,7,8,6)(10,12)(13,15,17,24)(14,20,21,28)(16,25,19,23)(18,22,26,27);(13,14,19,26,17,21,16,18)(15,22,25,28,24,27,23,20);(5,6)(7,8)(9,10)(11,12)(14,18)(16,19)(20,22)(21,26)(23,25)(27,28)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598808, #14278436, C₃×#10602270, …
maximal normal subgroup of	C₃×#10598808, #14278436, C₃×#10602270, …
maximal quotient of	C₃×#10602152, C₃×#10602089, C₃×#10602006, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4283);