C₅×#13646 #11936366

order	1280
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13646

Subgroups

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

202 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 41
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 41 points:

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

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, 6634);