C₁₃×#13198 #15699537

order	3328
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#13198

Subgroups

	subgroup	G/H
center	C₈×C₁₃	#127
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1955
prime radical	C₁₃×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

410 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 138 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)(14,15,18,23,19,24,34,45,20,25,35,46,36,47,60,71)(16,21,28,39,29,40,54,66,27,38,52,64,53,65,75,77)(17,22,32,43,33,44,49,62,31,42,56,68,59,70,74,72)(26,48,51,73,30,55,58,76,50,37,61,63,57,41,67,69);(14,16,18,28,19,29,34,54,20,27,35,52,36,53,60,75)(15,21,23,39,24,40,45,66,25,38,46,64,47,65,71,77)(17,30,56,67,33,50,74,51,31,57,32,58,59,26,49,61)(22,41,68,76,44,48,72,63,42,55,43,69,70,37,62,73);(14,17)(15,22)(16,26,27,50)(18,32)(19,33)(20,31)(21,37,38,48)(23,43)(24,44)(25,42)(28,51,52,61)(29,30,53,57)(34,49)(35,56)(36,59)(39,63,64,73)(40,41,65,55)(45,62)(46,68)(47,70)(54,58,75,67)(60,74)(66,69,77,76)(71,72)

Conjugacy classes & character table

676 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6186);