C₅×#67553 #14279477

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

Aliases

direct product

C₅×#67553

Splittings

Hall	C₃×C₅ ⋊ #1411
Sylow	C₃ ⋊ C₅×#1411

Subgroups

	subgroup	G/H
center	C₂×C₅	#4474
commutator	C₃×C₄²	C₂×C₄×C₅
Frattini	C₄⋊Q₈	C₂×C₅×D₃
Fitting	C₃×C₅×#356	C₂
socle	C₂×C₃×C₅	#353
prime radical	C₂²×C₃×C₅	#99
cosocle	C₃×C₄⋊Q₈	C₂²×C₅
hypercenter	C₅×#356	D₃
hypocenter	C₃	C₅×#1411
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

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

normal subgroups only

This group has 188 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)(9,11,14,23)(10,15,19,32)(12,24,26,37)(13,27,29,22)(16,31,18,20)(17,34,36,39)(21,25,38,28)(30,35,40,33);(6,7,8)(9,10,12,17,14,19,26,36)(11,20,21,34,23,31,38,39)(13,18,28,35,29,16,25,33)(15,24,30,27,32,37,40,22)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 5175);