C₅₃×#13723 #27221235

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅₃×#13723

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
prime radical	C₅₃×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2)(3,5)(4,6)(7,8)(10,11)(12,13)(14,25)(15,30)(17,27)(18,22)(19,35)(20,34)(21,24)(23,36)(26,33)(28,29)(31,39)(32,38)(37,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,4,7)(2,5,6,8)(9,10,17,32,14,19,26,23,30,36,29,35,39,38,12,11,16,21,13,18,31,37,28,34,15,20,33,40,25,22,27,24);(2,6)(5,8)(9,11,12,22,25,35,29,20,15,23,26,37,31,32,17,21,16,24,27,38,39,40,33,36,30,34,28,19,14,18,13,10)

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6711);