C₅₃×#10379 #27217891

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#10379

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

360 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,16)(14,15)(17,18)(19,26,28,32)(20,24)(21,30)(22,25)(23,29,31,27)(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);(4,13)(6,15)(7,12)(10,14)(19,27,28,29)(23,26,31,32);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,19)(18,23)(20,27)(21,29)(22,28)(24,32)(25,31)(26,30)

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, 3367);