Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #12877 |
366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(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)(54,57,59,82,60,72,78,74)(55,62,81,86,63,87,65,91)(56,67,70,102,69,95,100,97)(58,77,79,73,80,111,112,104)(61,83,93,114,66,92,85,116)(64,90,107,103,88,106,75,71)(68,99,101,96,76,109,110,105)(84,115,98,94,89,117,108,113);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56)(55,61)(57,67)(58,76)(59,70)(60,69)(62,83)(63,66)(64,89)(65,85)(68,80)(71,94)(72,95)(73,105)(74,97)(75,98)(77,109)(78,100)(79,110)(81,93)(82,102)(84,88)(86,114)(87,92)(90,117)(91,116)(96,104)(99,111)(101,112)(103,113)(106,115)(107,108);(54,55)(56,66,69,61)(57,71)(58,75)(59,81)(60,63)(62,73)(64,79)(65,78)(67,94,95,113)(68,98,76,108)(70,85,100,93)(72,103)(74,106)(77,91)(80,107)(82,90)(83,96,92,105)(84,101,89,110)(86,111)(87,104)(88,112)(97,115,102,117)(99,114,109,116)Polycyclic presentation:
2^8,53|a^2=f,c^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh,i^c=i^52878 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85903);