Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C53 ⋊ #7441 |
136 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,3)(2,7,9,22)(4,16,17,29)(5,14)(6,15)(8,23)(10,21,19,27)(11,26)(12,18,30,32)(13,20)(24,31)(25,28)(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);(1,2)(3,11,13,23,15,28,14,31)(4,8,17,24)(5,19)(6,9)(7,12,21,29,22,30,27,16)(10,20)(18,26,32,25)(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)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,c^2=fgh,d^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,i^a=i^52,c^b=cf,d^b=defg,g^b=gh,i^b=i^52,d^c=dg,e^c=eh701 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57193);