C₃×#65375 #11749983

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#65375

Splittings

Fitting	C₃×C₅×C₄⋊Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #497
Sylow	C₅ ⋊ C₃×#497

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×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×D₂₀	C₂²
hypercenter	C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#497
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

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

normal subgroups only

This group has 248 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)(13,15,17,23)(14,19,18,27)(16,25,26,22)(20,28,24,21);(9,10,11,12)(14,18)(19,27)(20,24)(21,28);(5,6)(7,8)(10,12)(13,14)(15,21)(16,24)(17,18)(19,22)(20,26)(23,28)(25,27)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598754, #14279419, C₃×#10602037, …
maximal normal subgroup of	C₃×#10598754, #14279419, C₃×#10602037, …
maximal quotient of	C₃×#10602128, C₃×#10602130, C₃×#10602132, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4178);