C₅₃×#7526 #27215038

order	13568
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#7526

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1404
commutator	#125	C₂×C₄×C₅₃
Frattini, cosocle	#491	C₂²×C₅₃
prime radical	C₂²×C₅₃	AΣL(1,𝔽₁₆)
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄²×C₅₃ ◁ C₅₃×#491 ◁ G

126 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 85 points:

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

Conjugacy classes & character table

1007 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 514);