C₅×#4697 #11750196

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#4697

Splittings

Fitting	C₃×C₄×C₅×C₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #488
Sylow	C₃ ⋊ C₅×#488

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×D₁₂	C₂²
hypercenter	C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈

192 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,4,5)(6,7,8)(10,14)(13,15)(17,19,26,32,21,28,20,27)(18,23,29,31,22,30,24,25);(9,11,12,16)(10,13,14,15)(18,22)(23,30)(24,29)(25,31);(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,29)(21,22)(23,27)(24,26)(28,31)(30,32)

Conjugacy classes & character table

270 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₃×#488, C₂×C₅×#4697, D₅×#4697, …
maximal normal subgroup of	C₅×D₃×#488, C₂×C₅×#4697, D₅×#4697, …
maximal quotient of	C₅×#69763, C₅×#69767, C₅×#69768, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4391);