Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #12027 |
180 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 83 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)(54,57,60,80,62,82,61,81)(55,64,67,76,68,95,88,111)(56,70,73,104,75,77,74,105)(58,78,84,79,85,92,59,87)(63,89,90,117,91,93,114,109)(65,94,83,110,66,97,98,115)(69,99,100,101,96,116,108,112)(71,102,72,107,106,113,86,103);(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)(54,56,58,71,60,73,84,72,62,75,85,106,61,74,59,86)(55,63,65,69,67,90,83,100,68,91,66,96,88,114,98,108)(57,77,78,113,80,105,79,103,82,70,92,102,81,104,87,107)(64,93,94,116,76,109,110,112,95,89,97,99,111,117,115,101);(54,55,59,66,61,67,85,98,62,68,84,65,60,88,58,83)(56,69,72,90,74,100,71,91,75,96,86,114,73,108,106,63)(57,76,79,110,81,95,78,97,82,111,87,115,80,64,92,94)(70,101,103,89,105,116,113,117,77,112,107,93,104,99,102,109)Polycyclic presentation:
2^8,53|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,e^b=eh,i^b=i^52,e^c=eh884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 81228);