C₅₃×#13540 #27221052

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#13540

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#2099
prime radical	C₂³×C₅₃	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

234 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6528);