C₅×#33982 #11956702

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

Aliases

direct product

C₅×#33982

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

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

normal subgroups only

This group has 472 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)(8,14)(9,13)(10,27)(11,30)(12,18)(15,23)(16,24)(17,28)(19,35)(20,31)(21,49)(22,50)(25,42)(26,34)(29,56)(32,52)(33,51)(36,46)(37,47)(38,45)(39,55)(40,53)(41,54)(43,57)(44,67)(48,63)(58,68)(59,66)(60,64)(61,65)(62,69);(7,13)(9,23)(10,24)(11,28)(14,31)(15,16)(17,42)(18,34)(20,46)(21,47)(22,45)(25,55)(26,53)(27,32)(29,54)(30,33)(35,57)(36,37)(38,63)(39,56)(40,41)(43,64)(44,65)(48,69)(49,58)(50,59)(51,52)(60,61)(62,67)(66,68);(6,8)(7,14)(9,20)(10,21)(11,22)(12,19)(13,31)(15,36)(16,37)(17,38)(18,35)(23,46)(24,47)(25,48)(26,43)(27,49)(28,45)(29,44)(30,50)(32,58)(33,59)(34,57)(39,62)(40,60)(41,61)(42,63)(51,66)(52,68)(53,64)(54,65)(55,69)(56,67);(8,19)(14,35)(20,43)(21,44)(22,45)(31,57)(36,60)(37,61)(38,50)(46,64)(47,65)(48,66)(49,67)(58,62)(59,63)(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, 26970);