C₅×#83510 #14459949

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#83510

Splittings

Hall	C₃×C₅ ⋊ #3302
Sylow	C₃ ⋊ C₅×#3302

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting, prime radical	C₃×C₅×#547	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₅×#547	D₃
hypocenter	C₃	C₅×#3302
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×#547

772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 4 generators on 32 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 185647);

C5×#83510 #14459949