C₅×#68701 #14312422

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

Aliases

direct product

C₅×#68701

Splittings

Hall	C₃×C₅ ⋊ #2102
Sylow	C₃ ⋊ C₅×#2102

Subgroups

	subgroup	G/H
center	C₄×C₅	#875
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×H(3,ℤ/4)	C₂
socle	C₂×C₃×C₅	#400
prime radical	C₂³×C₃×C₅	#41
hypercenter	C₅×H(3,ℤ/4)	D₃
hypocenter	C₃	C₅×#2102
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×H(3,ℤ/4)

316 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 24
transitive degree	μ_t(G) = 240

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

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 38120);