C₅₃×#13502 #27221014

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13502
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13502

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2077
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

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, 6490);