C₅₃×#13722 #27221234

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅₃×#13722

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
prime radical	C₂×D₁₆×C₅₃	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

184 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 93 points:

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

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6710);