Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C53×#362 ⋊ C2 |
| Sylow | C53 ⋊ #13078 |
454 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 109 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)(55,63)(56,68)(58,78)(59,83)(60,74)(61,75)(62,87)(64,95)(65,72)(66,93)(67,96)(69,103)(70,100)(71,102)(73,81)(76,106)(77,99)(79,107)(80,110)(82,101)(84,92)(85,111)(86,91)(88,115)(89,114)(90,105)(94,113)(97,117)(98,116)(104,112)(108,109);(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)(54,56,58,77,60,71,80,85,61,70,81,69,59,82,79,104)(55,62,86,67,66,89,108,98,65,90,64,94,84,88,76,97)(57,68,78,99,74,102,110,111,75,100,73,103,83,101,107,112)(63,87,91,96,93,114,109,116,72,105,95,113,92,115,106,117);(54,55,59,66,61,65,60,84)(56,67,71,97,70,94,82,98)(57,72,74,93,75,63,83,92)(58,76,79,86,81,108,80,64)(62,85,88,69,90,104,89,77)(68,96,101,116,100,113,102,117)(73,106,107,95,78,109,110,91)(87,112,114,103,105,111,115,99)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh,i^c=i^52878 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 87034);