C₃×#65478 #11750086

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

Aliases

direct product

C₃×#65478

Splittings

Fitting	C₂×C₃×C₅×Q₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #470
Sylow	C₅ ⋊ C₃×#470

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×Q₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×D₅	D₄
hypercenter	C₂×C₃×Q₁₆	D₅
hypocenter	C₅	C₃×#470
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₁₆

220 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 82 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)(13,15,17,23)(14,19,18,26)(16,24,25,22)(20,28,27,21);(13,14,16,20,17,18,25,27)(15,21,22,26,23,28,24,19);(5,6)(7,8)(9,10)(11,12)(14,18)(19,26)(20,27)(21,28)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598811, #14278012, C₃×#10598812, …
maximal normal subgroup of	C₃×#10598811, #14278012, C₃×#10598812, …
maximal quotient of	C₃×#10602099, C₃×#10602102, C₃×#10602167, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4281);