C₁₃×#12112 #15698451

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12112

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#408 ◁ G

390 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 85
transitive degree	μ_t(G) = 1664

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,21)(22,23,29,35)(24,36,40,64)(25,41,44,61)(26,45,49,59)(27,34,28,33)(30,52,55,38)(31,50,57,43)(32,47,60,48)(37,65,69,82)(39,70,51,58)(42,73,76,56)(46,62,77,54)(53,74,81,85)(63,68,84,67)(66,71,79,80)(72,78,83,75);(14,15)(16,18)(17,19)(20,21)(22,24,29,40)(23,30,35,55)(25,37,44,69)(26,46,49,77)(27,39,28,51)(31,53,57,81)(32,58,60,70)(33,54,34,62)(36,59,64,45)(38,47,52,48)(41,71,61,80)(42,74,76,85)(43,67,50,68)(56,82,73,65)(63,83,84,72)(66,78,79,75);(14,16)(15,18)(17,20)(19,21)(22,25)(23,31)(24,37)(26,42)(27,50)(28,43)(29,44)(30,53)(32,56)(33,61)(34,41)(35,57)(36,63)(38,66)(39,68)(40,69)(45,72)(46,74)(47,78)(48,75)(49,76)(51,67)(52,79)(54,80)(55,81)(58,82)(59,83)(60,73)(62,71)(64,84)(65,70)(77,85)

Conjugacy classes & character table

403 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5100);