C₅₃×#13444 #27220956

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#13444

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#2054
prime radical	C₅₃×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

392 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,15)(4,17)(5,21)(6,25)(7,28)(8,14)(9,16)(10,19)(11,24)(12,27)(13,23)(18,32)(20,29)(22,31)(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)(2,9)(4,12)(5,22)(6,26)(7,29)(8,18)(10,27)(11,20)(13,31)(14,19)(15,21)(16,24)(17,23)(25,32)(28,30);(1,4,5,19,23,32,27,30,8,20,6,9,7,15,24,31)(2,10,11,17,28,22,25,3,14,16,12,29,13,26,21,18)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6432);