Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C53×#510 ⋊ C2 |
| Sylow | C53 ⋊ #13311 |
520 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 95 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)(58,75)(59,83)(60,76)(61,77)(62,84)(64,89)(65,94)(66,90)(67,74)(68,95)(70,99)(71,104)(72,100)(73,101)(78,106)(79,108)(80,111)(81,109)(82,110)(85,112)(86,88)(87,105)(91,98)(92,116)(93,107)(96,115)(97,114)(102,113)(103,117);(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)(54,56,61,73)(55,62,66,86)(57,69,77,101)(58,79,82,102)(59,72,60,71)(63,88,90,84)(64,91,93,96)(65,87,67,85)(68,92,97,78)(70,80,103,81)(74,105,94,112)(75,108,110,113)(76,104,83,100)(89,115,107,98)(95,106,114,116)(99,111,117,109);(54,55)(56,68)(57,74)(58,78)(59,67)(60,65)(61,66)(62,70)(63,76)(64,81)(69,98)(71,91)(72,96)(73,97)(75,107)(77,94)(79,85)(80,93)(82,92)(83,90)(84,108)(86,103)(87,102)(88,113)(89,110)(95,100)(99,105)(101,115)(104,114)(106,109)(111,116)(112,117)Polycyclic presentation:
2^8,53|b^2=h,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh,e^c=eh,i^c=i^52866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88305);