C₃×#65426 #11750034

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

Aliases

direct product

C₃×#65426

Splittings

Hall	C₃×C₅ ⋊ #470
Sylow	C₅ ⋊ C₃×#470

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×C₅	D₄
hypercenter	C₂×C₃×M₁₆	D₅
hypocenter	C₅	C₃×#470
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×M₁₆

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

normal subgroups only

This group has 220 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)(15,23)(18,27)(21,28)(24,26);(13,15,22,26,17,23,16,24)(14,18,25,28,20,27,19,21);(5,6)(7,8)(9,10)(11,12)(13,14,17,20)(15,21,23,28)(16,25,22,19)(18,26,27,24)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10597973, C₃×#10598027, #14278015, …
maximal normal subgroup of	C₃×#10597973, C₃×#10598027, #14278015, …
maximal quotient of	C₃×#10601664, C₃×#10601755, C₃×#10601667, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4229);