Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#1395 ⋊ C2 |
| Sylow | C53 ⋊ #13101 |
630 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 89 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)(54,57,61,79)(55,63,67,92)(56,70,74,105)(58,82,83,76)(59,78,60,77)(62,87,89,115)(64,93,80,90)(65,91,66,75)(68,96,99,113)(69,100,84,111)(71,106,81,102)(72,104,73,103)(85,112,107,117)(86,109,95,110)(88,108,94,114)(97,116,98,101);(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)(54,56,69,85)(55,62,86,98)(57,70,100,112)(58,81,60,72)(59,73,83,71)(61,74,84,107)(63,87,109,101)(64,68,66,88)(65,94,80,99)(67,89,95,97)(75,108,93,96)(76,106,78,103)(77,104,82,102)(79,105,111,117)(90,113,91,114)(92,115,110,116);(54,55,60,66,61,67,59,65)(56,68,73,98,74,99,72,97)(57,75,78,63,79,91,77,92)(58,80,69,95,83,64,84,86)(62,85,88,81,89,107,94,71)(70,101,104,96,105,116,103,113)(76,110,100,90,82,109,111,93)(87,106,114,117,115,102,108,112)Polycyclic presentation:
2^8,53|a^2=f,b^2=df,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,i^b=i^52,d^c=dh,e^c=eh,f^c=fh,i^c=i^52,f^d=fh710 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 87161);