C₅₃×#10123 #27217635

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#10123

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

442 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Conjugacy classes & character table

1643 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 3111);