C₅₃×#8126 #27215638

order	13568
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#8126

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#95
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

520 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 230 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:

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

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, 1114);