C₅₃×#10317 #27217829

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₃×#10317

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

554 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 81
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 81 points:

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

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, 3305);