C₅²×#473 #13103210

order	1600
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

direct product

C₅²×#473

Subgroups

	subgroup	G/H
center, socle	C₂×C₅²	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
prime radical	C₅²×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ C₂×C₄×C₅² ◁ G

400 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 42
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 42 points:

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

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, 5562);