C₅₃×#13532 #27221044

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13532

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#493
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#2093
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

312 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6520);