C₅₃×#8494 #27216006

order	13568
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#8494

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#379
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

732 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,11,14,16)(4,7)(5,9)(6,10)(8,13,15,12)(17,18)(19,26,28,22)(20,25)(21,24)(23,27,31,29)(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);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19)(18,22)(20,27)(21,28)(23,25)(24,26)(29,30)(31,32);(1,3,6,14)(2,8,10,15)(4,13,5,12)(7,11,9,16)(17,19)(18,23)(20,29)(21,28)(22,25)(24,31)(26,32)(27,30)

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, 1482);