C₅₃×#12299 #27219811

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#12299

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×M₁₆×C₅₃ ◁ G

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Conjugacy classes & character table

1325 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5287);