Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12872 |
398 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 398 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)(80,82,83,86)(81,85,84,87)(89,95)(91,104)(92,107)(94,108)(96,97)(98,113)(99,103)(100,115)(101,118)(105,106)(109,110)(111,119)(112,114)(116,117);(81,84)(85,87)(88,90,93,102)(89,94,99,112)(91,100,106,117)(92,101,107,118)(95,108,103,114)(96,109,113,119)(97,110,98,111)(104,115,105,116);(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)(80,81)(82,85)(83,84)(86,87)(88,89,92,97,93,99,107,98)(90,94,101,110,102,112,118,111)(91,103,105,96,106,95,104,113)(100,114,116,109,117,108,115,119)Polycyclic presentation:
2^8,79|a^2=f,b^2=h,c^2=d,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,i^a=i^78,e^c=egh,f^c=fh,g^c=gh2230 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85881);