C₅×#67572 #14278397

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#67572

Splittings

Hall	C₃×C₅ ⋊ #1358
Sylow	C₃ ⋊ C₅×#1358

Subgroups

	subgroup	G/H
center	C₂×C₅	#4485
commutator	C₃×C₄²	C₂×C₄×C₅
Frattini	C₄×C₈	C₂×C₅×D₃
Fitting	C₃×C₅×#364	C₂
socle	C₂×C₃×C₅	#331
prime radical	C₂²×C₃×C₅	#100
cosocle	C₃×C₄×C₈	C₂²×C₅
hypercenter	C₅×#364	D₃
hypocenter	C₃	C₅×#1358
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×#364

184 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 70 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)(7,8)(9,11,14,23)(10,15,19,31)(12,24,26,38)(13,27,28,22)(16,30,18,20)(17,33,35,37)(21,25,39,36)(29,32,40,34);(6,7,8)(9,10,12,17,13,16,25,32,14,19,26,35,28,18,36,34)(11,20,21,37,22,31,38,40,23,30,39,33,27,15,24,29)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 4095);