C₃×#65741 #11752274

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

Aliases

direct product

C₃×#65741

Splittings

Fitting	C₃×C₅×Q₁₆ ⋊ C₄
Hall	C₃×C₅ ⋊ #444
Sylow	C₅ ⋊ C₃×#444

Subgroups

	subgroup	G/H
center	C₂×C₃	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₃×C₄
Frattini	C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×Q₁₆	C₄
socle	C₂×C₃×C₅	C₄×D₄
prime radical	C₃×#4065	C₂
cosocle	C₄×D₅	C₂³×C₃
hypercenter	C₃×Q₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#444
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₁₆

208 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10604139, C₃×#10604138, #14315823, …
maximal normal subgroup of	C₃×#10604139, C₃×#10604138, #14315823, …
maximal quotient of	C₃×#10603969, C₃×#10604838, C₃×#10604062, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 6469);