C₃×#65336 #11749944

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

Aliases

direct product

C₃×#65336

Splittings

Fitting	C₃×C₅×C₂²⋊Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #452
Sylow	C₅ ⋊ C₃×#452

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×#709	C₂
hypercenter	C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₃×#452
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×C₂²⋊Q₈

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

normal subgroups only

This group has 304 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)(13,16,18,24)(14,20,21,28)(15,22,17,26)(19,27,25,23);(9,10)(11,12)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(5,6)(7,8)(13,14)(15,19)(16,23)(17,25)(18,21)(20,22)(24,27)(26,28)

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₅×#452, C₂×C₃×#65336, #14423518, …
maximal normal subgroup of	C₃×D₅×#452, C₂×C₃×#65336, #14423518, …
maximal quotient of	C₃×#10600498, C₃×#10600493, C₃×#10600510, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4139);