C₅×#13079 #11935799

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×#13079

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1893
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

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

normal subgroups only

This group has 486 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)(6,7,12,15,11,17,28,16)(8,18,14,27,21,32,20,10)(9,13,24,30,23,31,25,34)(19,36,29,22,37,35,33,26);(6,8,10,7,12,20,18,16,11,21,27,17,28,14,32,15)(9,19,26,13,25,29,35,30,23,37,22,31,24,33,36,34);(6,9)(7,13)(8,19)(10,26)(11,23)(12,25)(14,33)(15,34)(16,30)(17,31)(18,35)(20,29)(21,37)(22,27)(24,28)(32,36)

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, 6067);