C₃×#65409 #11750017

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#65409

Splittings

Hall	C₃×C₅ ⋊ #419
Sylow	C₅ ⋊ C₃×#419

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×D₂₀
commutator	C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×C₅	D₄
cosocle	C₂×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	D₅
hypocenter	C₅	C₃×#419
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×M₁₆

300 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 126 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)(9,11,12,10)(13,16,17,24,18,25,15,22)(14,20,19,23,21,28,26,27);(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,27,28,23);(5,6)(7,8)(9,10,12,11)(13,14)(15,19)(16,23)(17,26)(18,21)(20,24)(22,28)(25,27)

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10601208, C₃×#10600934, #14297402, …
maximal normal subgroup of	C₃×#10601208, C₃×#10600934, #14297402, …
maximal quotient of	C₃×#10601560, C₃×#10600649, C₃×#10600650, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4212);