Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9280 |
382 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 127 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)(80,82)(81,86)(83,91)(84,89)(85,93)(87,95)(88,92)(90,94)(96,99,101,112)(97,103,105,110)(98,107,109,123)(100,114,115,111)(102,116,117,126)(104,119,113,118)(106,120,121,127)(108,124,125,122);(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)(80,81,83,90,84,87,92,85)(82,86,91,94,89,95,88,93)(96,98,100,108,101,109,115,125)(97,102,104,106,105,117,113,121)(99,107,114,124,112,123,111,122)(103,116,119,120,110,126,118,127);(81,85,87,90)(82,88,89,91)(96,97)(98,106)(99,110,112,103)(100,113)(101,105)(102,108)(104,115)(107,120,123,127)(109,121)(111,118,114,119)(116,122,126,124)(117,125)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eh,f^c=fgh,i^c=i^781288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 66159);