Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C79 ⋊ #7216 |
276 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 2 generators on 119 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)(81,84,86,85)(88,90,92,100,93,101,104,114)(89,94,97,107,95,98,109,112)(91,99,103,113,102,105,115,116)(96,110,111,119,108,106,117,118);(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)(80,81)(82,84)(83,86)(85,87)(88,89,91,96,93,95,102,108)(90,98,99,110,101,94,105,106)(92,97,103,111,104,109,115,117)(100,112,113,119,114,107,116,118)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,d^b=dg,i^b=i^78,d^c=dh1288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56601);