C₅×C₁₆×Q₁₆ #11934772

order	1280
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×C₁₆×Q₁₆

Subgroups

	subgroup	G/H
center	C₂×C₅×C₁₆	D₄
commutator	C₄	C₂²×C₅×C₁₆
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle, prime radical	C₂²×C₅	C₈×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₁₆ ◁ C₄×C₅×C₁₆ ◁ G

218 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 37
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

560 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5040);