C₁₃×#12305 #15698644

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12305

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃
prime radical	C₂³×C₁₃	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

262 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 77
transitive degree	μ_t(G) = 832

A minimal faithful permutation representation — 3 generators on 77 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,19,39,20,25,41,24)(16,26,30,48,29,53,40,54)(17,31,34,47,33,22,45,61)(18,35,38,23,37,67,68,50)(21,36,51,28,44,58,43,59)(27,52,56,76,57,70,66,73)(32,62,64,49,65,74,72,46)(42,69,75,77,60,63,71,55);(14,16,20,29)(15,21,25,44)(17,27,33,57)(18,36,37,58)(19,40,41,30)(22,42,31,60)(23,48,50,54)(24,51,39,43)(26,35,53,67)(28,38,59,68)(32,63,65,69)(34,66,45,56)(46,73,49,76)(47,75,61,71)(52,62,70,74)(55,64,77,72);(14,17)(15,22,25,31)(16,27)(18,32)(19,34)(20,33)(21,42,44,60)(23,49,50,46)(24,47,39,61)(26,52,53,70)(28,55)(29,57)(30,56)(35,62,67,74)(36,63)(37,65)(38,64)(40,66)(41,45)(43,71,51,75)(48,76,54,73)(58,69)(59,77)(68,72)

Conjugacy classes & character table

325 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5293);