Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#2161 ⋊ C2 |
| Sylow | C53 ⋊ #12927 |
378 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 378 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)(56,69)(58,74)(59,76)(60,75)(61,77)(62,84)(64,88)(65,93)(66,73)(67,89)(68,94)(70,98)(71,100)(72,101)(78,104)(79,106)(80,108)(81,107)(82,109)(83,99)(85,111)(86,103)(87,112)(90,113)(91,114)(92,105)(95,97)(96,115)(102,117)(110,116);(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)(54,56,59,71,61,72,60,83)(55,62,66,86,67,87,65,85)(57,69,76,100,77,101,75,99)(58,79,80,102,82,110,81,70)(63,84,73,103,89,112,93,111)(64,90,78,68,92,95,91,96)(74,106,108,117,109,116,107,98)(88,113,104,94,105,97,114,115);(54,55,59,66,61,67,60,65)(56,68,71,95,72,96,83,90)(57,73,76,89,77,93,75,63)(58,78,80,92,82,91,81,64)(62,79,86,102,87,110,85,70)(69,97,100,115,101,113,99,94)(74,105,108,114,109,88,107,104)(84,98,103,106,112,117,111,116)Polycyclic presentation:
2^8,53|a^2=f,b^2=f,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh,e^c=eh,i^c=i^52878 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86111);