C₅×#1780⋊C₃ #14510684

order	1920
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product	C₅×#1780⋊C₃
unambiguous semidirect product	#1780⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#1780)⋊C₃

Splittings

Fitting, Hall	C₅×#1780 ⋊ C₃
Sylow	#1780 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₅	#120⋊C₃
commutator, hypocenter	C₂²×Q₈	C₃×C₄×C₅
Frattini	C₂²×C₄	C₂×C₅×A₄
Fitting	C₅×#1780	C₃
socle	C₂³×C₅	C₄×A₄
prime radical	C₅×#5948	C₂
cosocle	C₂²×#41	C₂×C₃×C₅
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₄×C₅

254 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 45
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5)(6,7)(8,13,9,11,12,10)(14,15,16,20,19,23,26,33)(17,34,25,31,29,37,39,44,18,21,38,32,28,36,41,43,27,22,24,42,30,35,40,45);(7,13)(10,11)(14,29,19,27)(15,36,23,34)(16,40,26,38)(17,18,28,30)(20,44,33,42)(21,22,35,37)(24,25,39,41)(31,32,43,45)

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, 236382);