C₅₃×#7832 #27215344

order	13568
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#7832

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

322 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 13568

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

(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27)(33,34,37,41)(35,38,43,47)(36,39,44,40)(42,45,48,46)(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)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26)(33,35,37,43)(34,38,41,47)(36,42,44,48)(39,45,40,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)(34,39)(36,44)(38,45)(40,41)(42,48)(46,47)

Conjugacy classes & character table

1961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 820);