C₅×#67785 #14278369

order	1920
isoclinism	isoclinic class of #67783
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×#67785

Splittings

Hall	C₃×C₅ ⋊ #1356
Sylow	C₃ ⋊ C₅×#1356

Subgroups

	subgroup	G/H
center	C₂×C₅	#4540
commutator	C₃×C₄²	C₂×C₄×C₅
Frattini	#97	C₂×C₅×D₃
Fitting	C₃×C₅×#491	C₂
socle	C₂×C₃×C₅	#329
prime radical	C₂²×C₃×C₅	#100
cosocle	C₃×#97	C₂²×C₅
hypercenter	C₅×#491	D₃
hypocenter	C₃	C₅×#1356
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#97 ◁ C₅×#491

156 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,3,4,5)(6,7,8)(9,11,14,23)(10,15,19,32)(12,24,25,29)(13,27,28,22)(16,31,18,20)(17,35,37,38)(21,26,39,33)(30,34,40,36);(7,8)(9,10,12,17,13,18,26,36,14,19,25,37,28,16,33,34)(11,20,21,38,22,15,29,30,23,31,39,35,27,32,24,40)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 4067);