C₅×#69173 #14313745

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#69173

Splittings

Fitting	C₃×C₅×#356 ⋊ C₂
Hall	C₃×C₅ ⋊ #2135
Sylow	C₃ ⋊ C₅×#2135

Subgroups

	subgroup	G/H
center	C₂×C₅	D₃×#99
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂×Q₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#356	C₂
socle	C₂×C₃×C₅	C₂×#99
prime radical	C₂×C₅×#239	C₂²
cosocle	C₂×C₃×Q₈	C₂³×C₅
hypercenter	C₅×#356	D₃
hypocenter	C₃	C₅×#2135
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₅×Q₈ ◁ C₅×#356

460 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 40 points:

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

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, 39443);