C₅₃×#10665 #27218177

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#10665

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#399
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

334 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2,4,7,6,3,5,8)(9,10)(11,15)(12,22,23,32)(13,17)(14,18)(16,24,31,25)(19,34,35,39)(20,28)(21,29)(26,33)(27,36,30,37)(38,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3)(2,6)(4,7)(5,8)(9,11,13,20,14,21,26,38)(10,15,17,28,18,29,33,40)(12,19,25,37,23,35,24,36)(16,30,32,34,31,27,22,39);(2,7)(3,8)(4,5)(9,12)(10,16)(11,19)(13,25)(14,23)(15,27)(17,32)(18,31)(20,37)(21,35)(22,33)(24,26)(28,39)(29,30)(34,40)(36,38)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 3653);