C₅₃×#13570 #27221082

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₃×#13570

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₅₃
Frattini, cosocle	#124	C₂³×C₅₃
prime radical	C₅₃×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#532 ◁ G

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

normal subgroups only

This group has 482 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:

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

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, 6558);