C₅×#12893 #11935613

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#12893

Subgroups

	subgroup	G/H
center	C₄×C₅	#427
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1835
prime radical	C₂³×C₅	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7)(8,20)(9,27,30,58)(10,32)(11,16)(12,35)(13,19)(14,23)(15,29,41,61)(17,21)(18,33)(22,50,53,36)(24,45)(25,37)(26,47)(28,59,49,42)(31,62,60,40)(34,43)(38,56)(39,48)(44,52,65,69)(46,55)(51,57,67,66)(54,63,68,64);(6,8,21,48,33,55,11,24,13,26,10,23,12,25,34,56)(7,14,16,20,35,45,17,37,19,39,43,47,18,38,32,46)(9,22,49,67,60,68,29,52,30,53,28,51,31,54,61,69)(15,36,40,44,59,64,27,57,41,50,62,65,42,63,58,66);(6,9,10,28,33,60,34,61,13,30,21,49,12,31,11,29)(7,15,16,40,35,59,17,27,19,41,43,62,18,42,32,58)(8,22,23,51,55,68,56,69,26,53,48,67,25,54,24,52)(14,36,20,44,45,64,37,57,39,50,47,65,38,63,46,66)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5881);