C₅₃×#62694 #27270206

order	13568
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₅₃ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₅₃×#62694

Subgroups

	subgroup	G/H
center	C₄²×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
socle	C₂²×C₅₃	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ G

6640 conjugacy classes of subgroups; only the 5670 normal subgroups are shown.

normal subgroups only

This group has 6640 conjugacy classes of subgroups — too many to draw. Only its 5670 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 — 6 generators on 85 points:

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

Conjugacy classes & character table

4028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 55682);