C₅×#67551 #14279511

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

Aliases

direct product

C₅×#67551

Splittings

Hall	C₃×C₅ ⋊ #1413
Sylow	C₃ ⋊ C₅×#1413

Subgroups

	subgroup	G/H
center	C₂×C₅	#4473
commutator	C₃×C₄²	C₂×C₄×C₅
Frattini	C₄⋊Q₈	C₂×C₅×D₃
Fitting	C₃×C₅×#356	C₂
socle	C₂×C₃×C₅	#354
prime radical	C₂²×C₃×C₅	#99
cosocle	C₃×C₄⋊Q₈	C₂²×C₅
hypercenter	C₅×#356	D₃
hypocenter	C₃	C₅×#1413
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×C₄⋊Q₈ ◁ C₅×#356

156 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 46 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)(9,11,14,21)(10,15)(12,24,26,30)(13,27,29,23)(16,20)(17,34)(18,37)(19,32)(22,25,39,28)(31,33)(35,40)(36,38);(6,7,8)(9,10,12,17,14,19,26,36)(11,20,22,38,21,37,39,34)(13,18,28,35,29,16,25,33)(15,30,31,23,32,24,40,27)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 5209);