C₅×#85417 #14511155

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#85417

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#557	D₃
prime radical	C₂×C₅×SL(2,𝔽₃)	C₂³
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

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) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236853);