Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #7227 |
248 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 79 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 — 2 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)(54,56,59,71,60,72,76,101,61,73,82,106,74,99,57,69)(55,62,65,85,95,105,63,68,67,87,96,114,66,86,81,98)(58,70,78,102,79,103,77,83,80,104,88,110,107,109,75,100)(64,90,92,116,115,113,89,111,94,84,108,112,93,117,91,97);(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)(54,55,58,64)(56,68,70,97)(57,65,75,92)(59,81,78,91)(60,66,79,93)(61,67,80,94)(62,83,84,101)(63,88,89,82)(69,87,100,90)(71,105,102,117)(72,85,103,112)(73,98,104,111)(74,95,107,115)(76,96,77,108)(86,110,113,106)(99,114,109,116)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,c^2=fgh,e^2=fg,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,i^a=i^52,d^b=dg,i^b=i^52,d^c=dh872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 56636);