C₅²×#479 #13103216

order	1600
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product

C₅²×#479

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₄²×C₅² ◁ G

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

normal subgroups only

This group has 360 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) = 34
transitive degree	μ_t(G) = 1600

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5568);