C₅×#71468 #14316094

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

Aliases

direct product

C₅×#71468

Splittings

Fitting	C₂×C₃×C₅×D₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #2186
Sylow	C₃ ⋊ C₅×#2186

Subgroups

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×D₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂×C₅×D₁₆	D₃
hypocenter	C₃	C₅×#2186
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×D₁₆

460 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 28
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4,5)(7,8)(10,12)(13,15,17,18)(14,20,21,26)(16,22,25,23)(19,24,27,28);(6,7,8)(13,14)(15,20)(16,24)(17,21)(18,26)(19,23)(22,27)(25,28);(9,10)(11,12)(14,19)(15,18)(16,23)(20,28)(21,27)(22,25)(24,26)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 41792);