C₅₃×#12068 #27219580

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#12068

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#442
commutator	C₈	C₂²×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1580
prime radical	C₅₃×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₅₃×#346 ◁ G

204 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5056);