C₅₃×#10669 #27218181

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#10669

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

296 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 3392

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

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

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, 3657);