C₅₃×#11486 #27218998

order	13568
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#11486

Subgroups

	subgroup	G/H
center	C₄×C₅₃	C₂×#99
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×M₁₆	C₂³×C₅₃
socle	C₂×C₅₃	#1459
prime radical	C₅₃×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

364 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

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, 4474);