C₅×#13076 #11935796

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13076

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1893
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3060	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

466 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 146 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)(6,7,11,18,13,17,12,33)(8,19,22,25,21,37,23,29)(9,24,26,36,27,15,14,34)(10,28,30,35,32,20,31,16);(6,8,10,15,11,22,30,34,13,21,32,24,12,23,31,36)(7,14,35,37,33,27,28,25,17,26,16,19,18,9,20,29);(6,9)(7,15)(8,20)(10,29)(11,14)(12,26)(13,27)(16,23)(17,24)(18,36)(19,31)(21,28)(22,35)(25,32)(30,37)(33,34)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6064);