Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12657 |
334 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(84,87,88,89,90,102,103,105)(85,92,93,104,95,109,110,94)(86,97,112,113,99,100,98,101)(91,106,111,114,107,115,96,108);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(81,83)(84,86,89,100,88,98,102,113,90,99,105,97,103,112,87,101)(85,91,94,106,93,96,92,108,95,107,104,115,110,111,109,114);(80,81,82,83)(84,85)(86,96,99,111)(87,92)(88,93)(89,104)(90,95)(91,98,107,112)(94,105)(97,108,100,114)(101,115,113,106)(102,109)(103,110)Polycyclic presentation:
2^8,79|a^2=e,b^2=f,c^2=de,d^2=h,f^2=g,g^2=h,b^a=bd,f^a=fh,c^b=cg,e^b=eh,f^c=fh,i^c=i^781780 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 84893);