C₅₃×#7477 #27214989

order	13568
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7477

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#357
commutator	C₁₆	C₄²×C₅₃
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅₃
prime radical	C₂³×C₅₃	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

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

normal subgroups only

This group has 178 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) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 89 points:

(1,2,3,4)(5,6)(7,19)(8,15)(9,22)(10,28)(11,31)(12,18)(13,20)(14,25)(16,24)(17,29)(21,35)(23,32)(26,36)(27,33)(30,34)(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);(5,7,9,17,10,21,26,28,29,14,32,6,13,15,8,16,12,23,25,31,30,36,35,34,11,22,19,18,24,33,27,20)

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 465);