C₅×#3028⋊C₃ #14510986

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

Aliases

direct product	C₅×#3028⋊C₃
unambiguous semidirect product	#3028⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3028)⋊C₃

Splittings

Fitting, Hall	C₅×#3028 ⋊ C₃
Sylow	#3028 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×#120⋊C₃
commutator, cosocle, hypocenter	H(5,𝔽₂)	C₂²×C₃×C₅
Frattini	C₂³	C₂²×C₅×A₄
Fitting	C₅×#3028	C₃
prime radical	C₅×#5950	C₂
hypercenter	C₄×C₅	#120⋊C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅

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

normal subgroups only

This group has 312 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236684);