C₅₃×#13680 #27221192

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13677
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#13680

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2196
commutator, Frattini, cosocle	#124	C₂³×C₅₃
prime radical	C₅₃×#3289	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ #124×C₅₃ ◁ G

378 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Conjugacy classes & character table

1166 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6668);