C₅₃×#7046 #27214558

order	13568
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7046

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅₃	#95
commutator	C₂²×C₄	C₄²×C₅₃
Frattini, cosocle	C₂²×C₄²	C₂²×C₅₃
prime radical	C₂⁴×C₅₃	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

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

normal subgroups only

This group has 364 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 — 2 generators on 85 points:

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

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 34);