C₃×#66006 #11755683

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#66006

Splittings

Hall	C₃×C₅ ⋊ #563
Sylow	C₅ ⋊ C₃×#563

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₃×#124	D₅
hypocenter	C₅	C₃×#563
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

392 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 4 generators on 32 points:

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

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10602121, C₃×#10599147, #14290955, …
maximal normal subgroup of	C₃×#10602121, C₃×#10599147, #14290955, …
maximal quotient of	C₃×#10608040, C₃×#10607222, C₃×#10614688, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 9878);