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 ⋊ #13006 |
472 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 472 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)(58,81)(59,78)(60,87)(61,79)(62,89)(64,98)(65,95)(66,100)(67,96)(68,102)(70,110)(71,108)(72,111)(73,109)(74,88)(75,83)(76,85)(77,82)(80,93)(84,106)(86,105)(90,113)(91,104)(92,117)(94,101)(97,116)(99,115)(103,114)(107,112);(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)(54,56,59,84,61,73,85,71)(55,62,88,90,67,92,65,99)(57,69,78,106,79,109,76,108)(58,72,82,70,83,112,60,86)(63,89,74,113,96,117,95,115)(64,97,101,103,80,91,66,68)(75,107,87,105,81,111,77,110)(93,104,100,102,98,116,94,114);(54,55,60,66,85,88,83,64,61,67,82,101,59,65,58,80)(56,68,72,62,84,91,70,99,73,103,112,92,71,97,86,90)(57,74,77,93,76,96,81,100,79,95,87,98,78,63,75,94)(69,104,107,113,106,114,105,89,109,116,111,115,108,102,110,117)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,d^2=g,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,i^c=i^52872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86586);