C₅₃×#10613 #27218125

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#10613

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 6784

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

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

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 3601);