C₅×#3459⋊C₃ #14511389

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product	C₅×#3459⋊C₃
unambiguous semidirect product	#3459⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3459)⋊C₃

Splittings

Fitting, Hall	C₅×#3459 ⋊ C₃
Sylow	#3459 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	C₂²⋊A₄
commutator, cosocle, hypocenter	#543	C₂×C₃×C₅
Frattini	C₂²	C₂×C₅×C₂²⋊A₄
Fitting	C₅×#3459	C₃
socle	C₂²×C₅	C₂×C₂²⋊A₄
LCS	G ▷ #543
UCS	C₁ ◁ C₂×C₄×C₅

500 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 29
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(7,10,8)(11,12,13)(14,15,18,21)(16,26,25,28,22,20,24,27,17,19,23,29);(8,13)(10,12);(6,12)(7,8)(9,10)(11,13)(14,22,18,23)(15,26,21,27)(16,17,24,25)(19,20,28,29)

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, 237087);