C₅₃×#8799 #27216311

order	13568
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅₃×#8799

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#382
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

220 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 101
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 101 points:

(1,2,6,10,7,12,5,11)(3,9,13,16,14,15,4,8)(17,18)(19,27,30,40)(20,31,33,24)(21,25)(22,26)(23,29,36,44)(28,41)(32,46,47,37)(34,38)(35,42)(39,43)(45,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,3,5,4,7,14,6,13)(2,8,10,9,12,16,11,15)(17,19)(18,23,26,36)(20,28)(21,29)(22,30)(24,35,31,45)(25,27,38,40)(32,42)(33,43)(34,44)(37,39,46,41)(47,48);(1,4,5,14,7,13,6,3)(2,9,10,16,12,15,11,8)(17,20,21,32,22,33,34,47)(18,24,25,37,26,31,38,46)(19,28,29,42,30,43,44,48)(23,35,27,39,36,45,40,41)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1787);