C₅₃×#7495 #27215007

order	13568
nilpotency class	5
isoclinism	isoclinic class of #7493
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7495

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1390
commutator	C₂×D₄	C₄²×C₅₃
Frattini, cosocle	C₂×#124	C₂²×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#124×C₅₃ ◁ G

280 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 85 points:

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

Conjugacy classes & character table

1484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 483);