C₅×#68778 #14302558

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#68778

Splittings

Fitting	C₃×C₅×#339 ⋊ C₂
Hall	C₃×C₅ ⋊ #1906
Sylow	C₃ ⋊ C₅×#1906

Subgroups

	subgroup	G/H
center	C₄×C₅	#895
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#339	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₅×#5748	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#339	D₃
hypocenter	C₃	C₅×#1906
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#339

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

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, 28256);

C5×#68778 #14302558