C₁₃×#13199 #15699538

order	3328
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#13199

Subgroups

	subgroup	G/H
center	C₈×C₁₃	#124
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

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

normal subgroups only

This group has 406 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)(14,15,19,25,20,26,41,55,21,27,42,56,18,24,29,44)(16,28,33,57,31,46,58,54,32,52,40,68,38,22,43,47)(17,23,36,50,35,49,64,73,37,51,67,76,39,53,59,70)(30,60,63,75,34,45,65,72,62,48,69,74,61,71,66,77);(14,16,29,58,18,38,42,33,21,32,41,43,20,31,19,40)(15,22,44,57,24,52,56,47,27,46,55,68,26,28,25,54)(17,34,64,63,39,30,36,66,37,61,59,69,35,62,67,65)(23,48,73,72,53,45,50,75,51,60,70,77,49,71,76,74);(14,17)(15,23)(16,30,32,62)(18,39)(19,36)(20,35)(21,37)(22,45,46,71)(24,53)(25,50)(26,49)(27,51)(28,48,52,60)(29,59)(31,34,38,61)(33,63,40,69)(41,64)(42,67)(43,66,58,65)(44,70)(47,72,54,77)(55,73)(56,76)(57,74,68,75)

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, 6187);