C₅₃×#7336 #27214848

order	13568
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#7336

Subgroups

	subgroup	G/H
center	C₈×C₅₃	#99
commutator	C₂³	C₂×C₁₆×C₅₃
Frattini, cosocle	C₂×M₃₂	C₂²×C₅₃
socle	C₂×C₅₃	#1314
prime radical	C₂⁴×C₅₃	C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×M₃₂×C₅₃ ◁ G

170 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324);