Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #8504 |
728 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 131 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 — 3 generators on 119 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(82,87)(84,86)(88,91)(89,95)(90,99)(92,106)(93,103)(94,108)(96,110)(97,113)(98,114)(100,118)(101,116)(102,111)(104,107)(105,117)(109,115)(112,119);(80,81)(82,86)(83,85)(84,87)(88,90)(89,94)(91,99)(92,105)(93,101)(95,108)(96,98)(97,112)(100,107)(102,115)(103,116)(104,118)(106,117)(109,111)(110,114)(113,119);(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)(81,84,85,86)(88,89)(90,98,101,94)(91,102,103,113)(92,97)(93,96)(95,104,110,106)(99,115)(100,112,105,109)(107,111)(108,117)(114,118)(116,119)Polycyclic presentation:
2^8,79|a^2=f,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,i^a=i^78,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fg1522 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 62001);