C₅×#3488⋊C₃ #14511345

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product	C₅×#3488⋊C₃
unambiguous semidirect product	#3488⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3488)⋊C₃

Splittings

Fitting, Hall	C₅×#3488 ⋊ C₃
Sylow	#3488 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂³×A₄
commutator, cosocle	C₂×Q₈	C₂³×C₃×C₅
Frattini	C₂²	C₂³×C₅×A₄
Fitting	C₅×#3488	C₃
hypercenter	C₅×C₂²⋊Q₈	A₄
hypocenter	Q₈	C₂×C₃×C₅×D₄
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊Q₈

766 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 33
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2,3,4,5)(10,20,30,15,13,28,16,22,21,12,23,31)(11,25,33,14,18,32,19,24,26,17,27,29);(7,9)(10,19,16,11)(12,14,15,17)(13,25,23,24)(18,20,27,22)(21,28,30,31)(26,29,33,32);(6,7)(8,9)(10,11,16,19)(12,14,15,17)(13,18,23,27)(20,24,22,25)(21,26,30,33)(28,29,31,32)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237043);