C₅×#2889⋊C₃ #14510958

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product	C₅×#2889⋊C₃
unambiguous semidirect product	#2889⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#2889)⋊C₃

Splittings

Fitting, Hall	C₅×#2889 ⋊ C₃
Sylow	#2889 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	A₄×C₂²⋊C₄
commutator	C₂×Q₈	C₂×C₃×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×A₄
Fitting	C₅×#2889	C₃
prime radical	C₂×C₅×2₋¹⁺⁴⋊C₃	C₂
cosocle	C₂×#41	C₂²×C₃×C₅
hypercenter	C₅×#100	A₄
hypocenter	Q₈	C₃×C₅×C₂²⋊C₄
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#100

368 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 68 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)(7,12)(8,16)(9,19,10)(14,29,15,24,28,25)(17,33,18,30,32,31)(20,22,21)(26,36,27)(34,37,35);(6,28,16,37,11,29,8,36)(7,32,23,19,12,33,13,20)(9,15,30,35,21,25,17,27)(10,24,31,26,22,14,18,34)

Conjugacy classes & character table

215 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236656);