C₅×#3497⋊C₃ #14511347

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

solvable

Aliases

direct product	C₅×#3497⋊C₃
unambiguous semidirect product	#3497⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3497)⋊C₃

Splittings

Fitting, Hall	C₅×#3497 ⋊ C₃
Sylow	#3497 ⋊ 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×#3497	C₃
hypercenter	C₅×#121	A₄
hypocenter	Q₈	C₂×C₃×C₅×D₄
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#121

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

normal subgroups only

This group has 744 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) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 25 points:

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

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, 237045);