C₅×#69296 #14318001

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#69296

Splittings

Hall	C₃×C₅ ⋊ #2227
Sylow	C₃ ⋊ C₅×#2227

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₅×C₄⋊C₁₆	D₃
hypocenter	C₃	C₅×#2227
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₅×C₄⋊C₁₆

260 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 1920

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 43699);