C₅²×#464 #13103201

order	1600
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

direct product

C₅²×#464

Subgroups

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

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

normal subgroups only

This group has 448 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)(12,14)(15,16)(17,23)(18,20)(19,27)(21,25)(22,26)(24,30)(28,29);(6,7,8,9,10)(11,12,13,14)(16,20)(21,29)(22,27)(23,30);(15,17,18,24)(16,21,20,29)(19,28,26,25)(22,30,27,23)

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, 5553);