Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#1413 ⋊ C2 |
| Sylow | C53 ⋊ #13612 |
304 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(54,57)(55,62)(56,68)(58,74)(59,75)(60,73)(61,82)(63,89)(64,87)(65,86)(66,88)(67,94)(69,95)(70,96)(71,72)(76,107)(77,105)(78,102)(79,101)(80,106)(81,103)(83,108)(84,99)(85,109)(90,114)(91,113)(92,112)(93,115)(97,104)(98,100)(110,116)(111,117);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,56,60,71)(55,61,66,85)(57,72,73,68)(58,76,78,97)(59,79,81,70)(62,82,88,109)(63,84,65,67)(64,91,93,110)(69,77,98,80)(74,104,102,107)(75,96,103,101)(83,92,111,90)(86,94,89,99)(87,113,115,116)(95,106,100,105)(108,112,117,114);(54,55,58,64,60,66,78,93)(56,67,69,91,71,84,98,110)(57,62,74,87,73,88,102,115)(59,65,80,92,81,63,77,90)(61,76,83,79,85,97,111,70)(68,94,95,113,72,99,100,116)(75,86,106,112,103,89,105,114)(82,107,108,101,109,104,117,96)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=cg,d^b=dgh,e^b=efg,f^b=fh,i^b=i^52,d^c=dh,i^c=i^52,e^d=egh,f^e=fh389 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89988);