C₅×#1791⋊C₃ #14510686

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

Aliases

direct product	C₅×#1791⋊C₃
unambiguous semidirect product	#1791⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#1791)⋊C₃

Splittings

Fitting, Hall	C₅×#1791 ⋊ C₃
Sylow	#1791 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₅	#120⋊C₃
commutator, hypocenter	H(5,𝔽₂)	C₃×C₄×C₅
Frattini	C₂²×C₄	C₂×C₅×A₄
Fitting	C₅×#1791	C₃
socle	C₂×C₅	#379⋊C₃
prime radical	C₅×#5950	C₂
cosocle	#585	C₂×C₃×C₅
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₄×C₅

236 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 24 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)(6,7,12,25,11,16,8,13)(9,31,18,28,23,22,17,37,24,19,35,34,21,30,27,32,20,15,26,36,10,14,33,29);(6,20,11,23)(7,30,16,31)(8,33,12,35)(9,24,21,10)(13,36,25,37)(14,22,19,15)(17,27,26,18)(28,34,32,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236384);