C₁₃×#13656 #15699995

order	3328
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#13656

Subgroups

	subgroup	G/H
center	C₄×C₁₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃
socle	C₂×C₁₃	C₄×D₁₆
prime radical	C₁₃×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₈×C₁₃ ◁ C₄×C₈×C₁₃ ◁ C₄×C₁₃×C₁₆ ◁ G

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

normal subgroups only

This group has 184 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)(14,15,19,25,21,28,43,27)(16,29,32,60,34,61,33,67)(17,26,36,52,37,23,20,44)(18,38,40,70,42,51,75,71)(22,46,48,66,50,65,58,31)(24,35,54,68,56,69,55,41)(30,62,63,77,64,59,45,53)(39,57,72,76,74,47,73,49);(14,16)(15,22)(17,30)(18,39)(19,32)(20,45)(21,34)(23,47)(24,53)(25,48)(26,57)(27,58)(28,50)(29,38)(31,41)(33,43)(35,46)(36,63)(37,64)(40,72)(42,74)(44,49)(51,61)(52,76)(54,62)(55,59)(56,77)(60,70)(65,69)(66,68)(67,71)(73,75);(14,17,19,36,21,37,43,20)(15,23,25,44,28,26,27,52)(16,30,32,63,34,64,33,45)(18,35,40,68,42,69,75,41)(22,47,48,49,50,57,58,76)(24,51,54,71,56,38,55,70)(29,59,60,53,61,62,67,77)(31,39,46,72,66,74,65,73)

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, 6644);