C₅×#78716 #14425206

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

Aliases

direct product

C₅×#78716

Splittings

Fitting	C₃×C₅×#413 ⋊ C₂
Hall	C₃×C₅ ⋊ #3018
Sylow	C₃ ⋊ C₅×#3018

Subgroups

	subgroup	G/H
center	C₄×C₅	#882
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#413	C₂
socle	C₂×C₃×C₅	C₂×#120
hypercenter	C₅×#413	D₃
hypocenter	C₃	C₅×#3018
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#413

1448 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)(9,11,13,18)(10,15,16,22)(12,17,19,23)(14,21,20,24);(12,19)(14,20)(17,23)(21,24);(7,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,24)(18,22)(21,23);(7,8)(10,14)(12,19)(15,21)(16,20)(17,23)(22,24)

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, 150904);

C5×#78716 #14425206