C₅₃×#13488 #27221000

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13488

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

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

normal subgroups only

This group has 302 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) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 points:

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

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, 6476);