C₅²×#470 #13103207

order	1600
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

direct product

C₅²×#470

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
prime radical	C₂³×C₅²	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₂²×C₄×C₅² ◁ G

440 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 30
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5559);