C₅×#71502 #14316196

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

Aliases

direct product

C₅×#71502

Splittings

Fitting	C₂×C₃×C₅×Q₃₂ ⋊ C₂
Hall	C₃×C₅ ⋊ #2188
Sylow	C₃ ⋊ C₅×#2188

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×Q₃₂	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂²×C₅×D₃	D₈
hypercenter	C₂×C₅×Q₃₂	D₃
hypocenter	C₃	C₅×#2188
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×Q₃₂

432 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 44
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,3,4,5)(7,8)(10,12);(6,7,8)(13,15,18,26)(14,19,22,36)(16,28,31,44)(17,33,29,25)(20,38,39,43)(21,40,32,35)(23,41,42,27)(24,37,34,30);(9,10)(11,12)(13,14,18,22)(15,23,26,42)(16,27,31,41)(17,32,29,21)(19,34,36,24)(20,37,39,30)(25,43,33,38)(28,40,44,35)

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, 41894);