C₅₃×#13472 #27220984

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13472

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2067
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

316 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

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, 6460);