C₅×#78690 #14425667

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#78690

Splittings

Fitting	C₃×C₅×#410 ⋊ C₂
Hall	C₃×C₅ ⋊ #3025
Sylow	C₃ ⋊ C₅×#3025

Subgroups

	subgroup	G/H
center	C₄×C₅	#882
commutator, cosocle	C₂³×C₃	C₂⁴×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	C₂×#120
hypercenter	C₅×#410	D₃
hypocenter	C₃	C₅×#3025
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#410

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3,4,5)(6,7,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24);(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(7,8)(14,20)(21,24)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 151365);

C5×#78690 #14425667