C₅×#67580 #14279622

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

Aliases

direct product

C₅×#67580

Splittings

Fitting	C₃×C₅×#367 ⋊ C₂
Hall	C₃×C₅ ⋊ #1418
Sylow	C₃ ⋊ C₅×#1418

Subgroups

	subgroup	G/H
center	C₂×C₅	#4491
commutator	C₃×C₁₆	C₂×C₄×C₅
Frattini	C₂×C₁₆	C₂×C₅×D₃
Fitting	C₃×C₅×#367	C₂
socle	C₂×C₃×C₅	#357
prime radical	C₅×#4904	C₂
cosocle	C₂×C₃×C₁₆	C₂²×C₅
hypercenter	C₅×#367	D₃
hypocenter	C₃	C₅×#1418
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×C₁₆ ◁ C₅×#367

148 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 62 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)(9,10)(11,20)(12,25)(13,28)(14,16)(15,29)(17,24)(18,26)(19,31)(21,34)(22,35)(23,32)(27,30)(33,38)(36,40)(37,39);(6,7,8)(10,15,16,31)(11,19)(12,24)(13,26)(17,35,34,39)(18,36,28,33)(20,25,32,30)(21,27)(22,40)(23,29)(37,38)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 5320);