Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #13544 |
212 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 81 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)(54,57,58,74,59,75,61,83)(55,63,64,88,67,90,66,97)(56,69,70,100,71,101,82,102)(60,80,77,108,78,99,81,76)(62,84,68,98,86,112,96,111)(65,93,91,115,95,117,94,89)(72,104,79,73,103,106,109,107)(85,113,92,87,110,105,116,114);(54,56,61,82,59,71,58,70)(55,62,64,68,67,86,66,96)(57,73,83,104,75,107,74,106)(60,79,81,72,78,109,77,103)(63,87,88,105,90,114,97,113)(65,92,91,110,95,116,94,85)(69,99,102,108,101,80,100,76)(84,93,98,115,112,117,111,89);(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)(54,55)(56,68,71,96)(57,63,75,90)(58,64)(59,67)(60,65)(61,66)(62,70,86,82)(69,98)(72,92,103,116)(73,105)(74,88,83,97)(76,89,108,115)(77,91)(78,95)(79,110,109,85)(80,93,99,117)(81,94)(84,102)(87,104)(100,112)(101,111)(106,114)(107,113)Polycyclic presentation:
2^8,53|a^2=dg,b^2=de,c^2=d,d^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^52,c^b=cf,f^b=fg,g^b=gh,f^c=fg,g^c=gh,i^c=i^52872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89726);