Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #12839 |
362 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,57)(55,62)(56,68)(58,73)(59,75)(60,74)(61,80)(63,83)(64,72)(65,84)(66,86)(67,91)(69,94)(70,96)(71,95)(76,106)(77,104)(78,105)(79,102)(81,100)(82,108)(85,101)(87,110)(88,93)(89,111)(90,112)(92,113)(97,115)(98,116)(99,114)(103,107)(109,117);(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)(54,56,67,90,59,70,78,98,60,71,92,109,77,97,58,69)(55,61,76,89,65,82,85,88,64,81,107,99,79,87,63,66)(57,68,91,112,75,96,105,116,74,95,113,117,104,115,73,94)(62,80,106,111,84,108,101,93,72,100,103,114,102,110,83,86);(54,55)(56,66)(57,72,74,62)(58,76)(59,79)(60,64)(61,69)(63,67)(65,77)(68,93,95,86)(70,99)(71,88)(73,103,105,106)(75,84,104,102)(78,107)(80,94,100,116)(81,98)(82,109)(83,91,101,113)(85,92)(87,90)(89,97)(96,111,115,114)(108,117,110,112)Polycyclic presentation:
2^8,53|a^2=f,b^2=dg,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,i^b=i^52,f^c=fh884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85755);