C₅×#3458⋊C₃ #14511388

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

solvable

Aliases

direct product	C₅×#3458⋊C₃
unambiguous semidirect product	#3458⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3458)⋊C₃

Splittings

Fitting, Hall	C₅×#3458 ⋊ C₃
Sylow	#3458 ⋊ 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×#3458	C₃
socle	C₂²×C₅	C₂×C₂²⋊A₄
prime radical	C₅×#543⋊C₃	C₂
LCS	G ▷ #543
UCS	C₁ ◁ C₂×C₄×C₅

432 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 50 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)(7,10,12)(9,11,13)(14,15,18,21)(16,26,17,27,22,29,23,28,25,19,24,20);(6,8)(7,11)(9,12)(10,13)(14,18)(15,21)(16,23)(19,27);(6,10,8,13)(7,12,11,9)(14,24,18,22)(15,28,21,26)(16,25,23,17)(19,29,27,20)

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, 237086);