C₅×#67691 #14279076

order	1920
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#67691

Splittings

Hall	C₃×C₅ ⋊ #1391
Sylow	C₃ ⋊ C₅×#1391

Subgroups

	subgroup	G/H
center	C₄×C₅	#834
commutator	C₂³×C₃	C₄²×C₅
Frattini	C₂×#41	C₂×C₅×D₃
Fitting	C₃×C₅×#408	C₂
socle	C₂×C₃×C₅	#342
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂×C₃×#41	C₂²×C₅
hypercenter	C₅×#408	D₃
hypocenter	C₃	C₅×#1391
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ C₅×#408

332 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 106 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)(9,11,13,21,14,22,25,35)(10,15,18,28,16,19,30,33)(12,20,24,34,23,26,36,37)(17,31,32,40,29,27,38,39);(7,8)(9,10,12,17)(11,19,20,31)(13,18,24,32)(14,16,23,29)(15,26,27,22)(21,33,34,40)(25,30,36,38)(28,37,39,35)

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, 4774);