C₅₃×#13511 #27221023

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13459
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#13511

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#2084
prime radical	C₂³×C₅₃	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

250 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 110 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:

(1,2,6,11,5,10,19,7)(3,12,15,23,14,8,16,25)(4,17,20,27,21,28,22,9)(13,30,31,24,26,18,32,29)(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);(1,3,5,14)(2,8,10,12)(4,18)(6,15,19,16)(7,23,11,25)(9,26)(13,27)(17,32)(20,29)(21,30)(22,24)(28,31);(1,4,19,20,5,21,6,22)(2,9,7,17,10,27,11,28)(3,13,15,32,14,26,16,31)(8,24,25,30,12,29,23,18)

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, 6499);