C₁₃×#12302 #15698641

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12302

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

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

normal subgroups only

This group has 310 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)(14,15,20,25,21,27,19,26)(16,28,32,49,33,57,31,56)(17,34,36,23,37,52,42,48)(18,38,40,24,41,53,43,51)(22,39,54,30,46,62,45,63)(29,58,60,55,61,76,64,73)(35,50,69,66,70,47,68,71)(44,72,65,67,75,59,74,77);(14,16,21,33)(15,22,27,46)(17,29,37,61)(18,39,41,62)(19,32,20,31)(23,44,48,75)(24,49,51,56)(25,45,26,54)(28,38,57,53)(30,40,63,43)(34,65,52,74)(35,67,70,77)(36,64,42,60)(47,58,50,76)(55,71,73,66)(59,69,72,68);(14,17)(15,23)(16,29)(18,35)(19,42)(20,36)(21,37)(22,44)(24,50)(25,52)(26,34)(27,48)(28,55)(30,59)(31,64)(32,60)(33,61)(38,71)(39,67)(40,69)(41,70)(43,68)(45,74)(46,75)(47,51)(49,76)(53,66)(54,65)(56,58)(57,73)(62,77)(63,72)

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, 5290);