C₁₃×#13078 #15699417

order	3328
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₃×#13078

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1893
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

454 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 146 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)(14,15,19,26,21,25,20,44)(16,27,31,57,30,54,42,58)(17,32,34,53,35,23,43,52)(18,36,39,46,41,68,40,24)(22,45,48,29,50,64,49,37)(28,56,61,76,60,73,62,77)(33,66,67,55,38,69,70,51)(47,72,74,63,65,71,75,59);(14,16,18,37,20,31,40,45,21,30,41,29,19,42,39,64)(15,22,46,27,26,49,68,58,25,50,24,54,44,48,36,57)(17,28,38,59,34,62,70,71,35,60,33,63,43,61,67,72)(23,47,51,56,53,74,69,76,32,65,55,73,52,75,66,77);(14,17)(15,23)(16,28)(18,38)(19,43)(20,34)(21,35)(22,47)(24,55)(25,32)(26,53)(27,56)(29,63)(30,60)(31,62)(33,41)(36,66)(37,59)(39,67)(40,70)(42,61)(44,52)(45,71)(46,51)(48,75)(49,74)(50,65)(54,73)(57,77)(58,76)(64,72)(68,69)

Conjugacy classes & character table

598 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6066);