C₅×#13717 #11936437

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13717
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13717

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#2218	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂ ◁ G

170 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 160

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

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6705);