Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #8791 |
262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(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)(80,83,84,92,85,93,91,82)(81,87,88,86,89,90,94,95)(96,99,100,111,101,112,113,126)(97,103,104,116,105,110,117,125)(98,107,108,121,109,122,123,127)(102,114,106,118,115,124,119,120);(80,82,91,93,85,92,84,83)(81,86,94,87,89,95,88,90)(96,98)(97,102,105,115)(99,107)(100,108)(101,109)(103,114,110,124)(104,106,117,119)(111,121)(112,122)(113,123)(116,118,125,120)(126,127);(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)(82,90)(83,86)(84,94)(85,89)(87,92)(88,91)(93,95)(96,97)(98,106,109,119)(99,110,112,103)(100,104)(101,105)(102,108,115,123)(107,120)(111,125,126,116)(113,117)(114,121)(118,122)(124,127)Polycyclic presentation:
2^8,79|a^2=f,b^2=eg,c^2=d,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^78,f^b=fgh,f^c=fgh3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 63253);