C₃×#65787 #11752320

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

Aliases

direct product

C₃×#65787

Splittings

Hall	C₃×C₅ ⋊ #408
Sylow	C₅ ⋊ C₃×#408

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×Q₈	C₄
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂×C₃×#713	C₂
cosocle	C₂×Dic₅	C₂³×C₃
hypercenter	C₂×C₃×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#408
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈

316 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 104 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)(4,5,6,8,7)(10,12)(13,16,18,17)(14,20,22,21)(15,23,24,25)(19,26,27,28);(13,15,18,24)(14,19,22,27)(16,25,17,23)(20,28,21,26);(5,6,7,8)(9,10)(11,12)(13,14,17,21,18,22,16,20)(15,19,23,26,24,27,25,28)

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598578, #14279179, C₃×#10605020, …
maximal normal subgroup of	C₃×#10598578, #14279179, C₃×#10605020, …
maximal quotient of	C₃×#10604296, C₃×#10604978, C₃×#10604299, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 6515);