C₅₃×#11578 #27219090

order	13568
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#11578

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

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

normal subgroups only

This group has 426 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) = 3392

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

1961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 4566);