C₅×#13077 #11935797

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13077

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

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

normal subgroups only

This group has 470 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)(8,18,21,33)(9,14,24,32)(10,25)(11,17)(12,30)(13,20,31,26)(15,19)(16,29)(22,23,36,27)(28,37)(34,35);(6,8,19,32,29,36,10,26,11,21,35,14,12,22,28,20)(7,13,25,23,30,9,15,33,17,31,37,27,16,24,34,18);(6,9)(7,14)(8,15)(10,27)(11,24)(12,13)(16,26)(17,32)(18,35)(19,33)(20,30)(21,34)(22,25)(23,28)(29,31)(36,37)

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, 6065);