C₅×#69171 #14313673

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

Aliases

direct product

C₅×#69171

Splittings

Fitting	C₃×C₅×#356 ⋊ C₂
Hall	C₃×C₅ ⋊ #2133
Sylow	C₃ ⋊ C₅×#2133

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	C₄
cosocle	C₂×C₃×Q₈	C₂³×C₅
hypercenter	C₅×#356	D₃
hypocenter	C₃	C₅×#2133
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)(6,7)(11,20)(15,31)(19,33)(21,24)(22,27)(30,38)(32,40)(37,39);(7,8)(9,11)(10,15)(12,24)(13,27)(14,20)(16,33)(17,32)(18,31)(19,23)(21,26)(22,29)(25,30)(28,38)(34,39)(35,37)(36,40);(9,10,13,17,14,18,29,36)(11,19,22,37,20,33,27,39)(12,23,25,35,26,16,28,34)(15,30,32,24,31,38,40,21)

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, 39371);