Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #10045 |
444 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 10112 |
A minimal faithful permutation representation — 3 generators on 127 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)(81,85)(83,92)(84,89)(86,95)(87,94)(88,91)(90,93)(96,99,102,115)(97,104,107,122)(98,109,111,125)(100,114,101,116)(103,117,118,127)(105,121,106,123)(108,120,119,124)(110,113,112,126);(85,94)(88,95)(96,98,102,111)(97,103,107,118)(99,113,115,126)(100,112,101,110)(104,120,122,124)(105,119,106,108)(109,116,125,114)(117,123,127,121);(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)(82,88)(83,91)(84,87)(85,90)(86,93)(89,95)(92,94)(96,97,102,107)(98,108,111,119)(99,104,115,122)(100,106,101,105)(103,112,118,110)(109,124,125,120)(113,127,126,117)(114,123,116,121)Polycyclic presentation:
2^8,79|a^2=g,b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,i^a=i^78,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eh,f^c=fg1513 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 70617);