C₅×#33983 #11956703

order	1280
nilpotency class	5
isoclinism	isoclinic class of #33982
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#33983

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂²×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₅
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₅×#41 ◁ C₅×#131 ◁ C₅×#504 ◁ G

462 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 69 points:

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

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, 26971);