C₁₃×#13612 #15699951

order	3328
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#13612

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#2127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#536 ◁ G

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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)(14,15,18,24,20,26,38,53)(16,27,29,51,31,44,58,70)(17,22,34,47,33,48,62,75)(19,25,40,52,41,23,37,50)(21,36,43,39,45,57,71,30)(28,54,55,73,32,59,60,76)(35,46,66,72,63,49,65,74)(42,67,68,61,69,64,77,56);(14,16,20,31)(15,21,26,45)(17,32,33,28)(18,36,38,57)(19,39,41,30)(22,42,48,69)(23,44,25,27)(24,51,53,70)(29,37,58,40)(34,64,62,67)(35,56,63,61)(43,52,71,50)(46,54,49,59)(47,73,75,76)(55,66,60,65)(68,72,77,74);(14,17)(15,22)(16,28)(18,34)(19,35)(20,33)(21,42)(23,49)(24,47)(25,46)(26,48)(27,54)(29,55)(30,56)(31,32)(36,67)(37,65)(38,62)(39,61)(40,66)(41,63)(43,68)(44,59)(45,69)(50,74)(51,73)(52,72)(53,75)(57,64)(58,60)(70,76)(71,77)

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, 6600);