C₁₃×#33728 #15720067

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₃×#33728

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

482 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 53
transitive degree	μ_t(G) = 1664

A minimal faithful permutation representation — 4 generators on 53 points:

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

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, 26716);