C₅₃×#9500 #27217012

order	13568
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₃×#9500

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₂⁴	C₂²×C₄×C₅₃
Frattini, cosocle	C₂⁵	C₂³×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂⁵×C₅₃ ◁ G

660 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 85 points:

(1,2)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,31,32)(26,30)(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)(2,6,8,13)(4,9)(5,10)(7,12,14,16)(11,15)(17,19,21,26)(18,22,24,30)(20,25,28,31)(23,29,27,32);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32)

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, 2488);