Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9314 |
406 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 99 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)(82,90)(86,94)(89,95)(91,92)(96,99)(97,104)(98,107)(100,115)(101,112)(102,113)(103,117)(105,122)(106,123)(108,111)(109,126)(110,114)(116,125)(118,127)(119,124)(120,121);(80,82,85,90)(81,86)(83,91,84,92)(87,89)(88,94)(93,95)(96,98,100,109,102,108,101,116)(97,103,106,118,105,120,114,119)(99,111,115,125,113,107,112,126)(104,121,123,124,122,117,110,127);(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)(80,81,84,87,85,88,83,93)(82,89,92,94,90,95,91,86)(96,97,101,106,102,105,100,114)(98,103,109,119,108,120,116,118)(99,110,112,104,113,123,115,122)(107,124,126,121,111,127,125,117)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^78,c^b=cg,d^b=dh,f^b=fh,e^c=eh1753 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 66400);