C₅×#2785⋊C₃ #14510925

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₅ · Schur multiplier (H₂) = C₄

solvable

Aliases

direct product	C₅×#2785⋊C₃
unambiguous semidirect product	#2785⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#2785)⋊C₃

Splittings

Fitting, Hall	C₅×#2785 ⋊ C₃
Sylow	#2785 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	#5450
commutator, cosocle, hypocenter	C₂²×Q₈	C₂²×C₃×C₅
Frattini	C₂³	C₂²×C₅×A₄
Fitting	C₅×#2785	C₃
socle	C₂³×C₅	C₂²×A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂×C₅

216 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 120

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236623);