C₁₃×#32768 #15719107

order	3328
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₁₃×#32768

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×#117 ◁ G

736 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 61
transitive degree	μ_t(G) = 1664

A minimal faithful permutation representation — 4 generators on 61 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,18)(16,19,20,21)(22,23)(24,26)(25,27)(28,29)(30,31)(32,36)(33,37)(34,39)(35,38)(40,49)(41,51)(42,50)(43,52)(44,54)(45,53)(46,56)(47,57)(48,55)(58,60)(59,61);(15,18)(16,20)(22,24)(23,26)(25,28)(27,29)(30,32,35,42)(31,36,38,50)(33,43,45,59)(34,46,48,41)(37,52,53,61)(39,56,55,51)(40,47,58,44)(49,57,60,54);(16,20)(19,21)(23,27)(26,29)(30,33,35,45)(31,37,38,53)(32,40,42,58)(34,47,48,44)(36,49,50,60)(39,57,55,54)(41,59,46,43)(51,61,56,52);(14,16,17,20)(15,19,18,21)(24,28)(26,29)(31,38)(36,50)(37,53)(39,55)(49,60)(51,56)(52,61)(54,57)

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 25756);