C₅×#71477 #14317197

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

Aliases

direct product

C₅×#71477

Splittings

Fitting	C₂×C₃×C₅×SD₃₂ ⋊ C₂
Hall	C₃×C₅ ⋊ #2210
Sylow	C₃ ⋊ C₅×#2210

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×SD₃₂	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×#826	C₂
hypercenter	C₂×C₅×SD₃₂	D₃
hypocenter	C₃	C₅×#2210
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×SD₃₂

340 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,8)(14,19)(16,24)(17,23)(18,26)(20,21)(22,28)(25,27);(7,8)(9,10,11,12)(14,18)(19,26)(20,27)(21,25);(7,8)(10,12)(13,14)(15,18)(16,19)(17,25)(20,24)(21,23)(22,26)(27,28)

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, 42895);