C₅×#12897 #11935617

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12897

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	#432
prime radical	C₂³×C₅	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

278 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15,19,25,18,24,34,41,20,26,36,22,35,23,17,32)(16,27,30,42,29,21,28,37,31,38,43,45,44,39,33,40);(6,8,9,12)(7,10,11,13)(14,16,20,31)(15,21)(17,33,34,28)(18,29,35,44)(19,30,36,43)(22,40)(23,27)(24,38)(25,37)(26,39)(32,42)(41,45);(7,11)(10,13)(14,17,18,19,20,34,35,36)(15,22,23,41,26,25,24,32)(16,28,29,43,31,33,44,30)(21,37,27,42,39,40,38,45)

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5885);