Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #13157 |
296 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 93 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)(54,57,58,77,61,79,83,78,62,80,84,112,60,89,59,87)(55,64,65,97,67,99,81,75,68,100,85,111,90,98,66,103)(56,70,71,106,73,76,86,107,74,109,82,113,88,108,72,110)(63,91,69,104,94,96,102,114,95,105,101,117,93,116,92,115);(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)(54,56,58,82,61,73,83,72,62,74,84,71,60,88,59,86)(55,63,85,69,67,94,66,102,68,95,65,101,90,93,81,92)(57,76,112,107,79,109,87,113,80,108,77,110,89,70,78,106)(64,96,97,115,99,105,75,104,100,116,111,114,98,91,103,117);(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)(54,55,60,67,62,68,61,90)(56,69,73,92,74,101,88,102)(57,75,79,97,80,103,89,111)(58,81,59,85,84,66,83,65)(63,72,93,71,95,86,94,82)(64,77,98,78,100,112,99,87)(70,105,108,116,109,91,76,96)(104,113,114,107,117,106,115,110)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,i^a=i^52,c^b=cg,d^b=dh,e^b=eh,f^b=fh,i^b=i^52,e^c=eh,f^c=fh,i^c=i^52866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 87557);