C₅²×#446 #13103183

order	1600
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅² · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product

C₅²×#446

Subgroups

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

376 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 26
transitive degree	μ_t(G) = 1600

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

700 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5535);