C₁₃×#12466 #15698805

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃×#12466

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃
prime radical	C₁₃×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₄×C₈×C₁₃ ◁ G

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

normal subgroups only

This group has 186 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)(14,15,22,29)(16,30,37,62)(17,38,42,54)(18,26,43,48)(19,25,31,57)(20,28,21,27)(23,47,53,33)(24,39,56,40)(32,64,67,76)(34,51,44,52)(35,50,36,58)(41,71,46,59)(45,60,63,55)(49,70,74,66)(61,68,73,65)(69,75,72,77);(14,16,31,34,20,35,18,33,22,37,19,44,21,36,43,47)(15,23,48,50,27,51,25,30,29,53,26,58,28,52,57,62)(17,32,63,69,41,66,40,68,42,67,45,72,46,70,39,65)(24,49,71,75,55,64,38,61,56,74,59,77,60,76,54,73);(14,17)(15,24)(16,32)(18,40)(19,45)(20,41)(21,46)(22,42)(23,49)(25,38)(26,59)(27,55)(28,60)(29,56)(30,61)(31,63)(33,68)(34,69)(35,66)(36,70)(37,67)(39,43)(44,72)(47,65)(48,71)(50,75)(51,64)(52,76)(53,74)(54,57)(58,77)(62,73)

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, 5454);