Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #10630 |
272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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)(81,86)(82,88)(84,91)(85,92)(89,90)(93,95)(96,99,100,111,101,112,114,125)(97,103,104,117,105,118,120,127)(98,107,108,123,109,110,113,124)(102,115,106,121,116,126,119,122);(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)(80,82,83,89)(81,85,86,93)(84,88,91,90)(87,95,94,92)(96,98)(97,102)(99,110)(100,113)(101,109)(103,115)(104,119)(105,116)(106,120)(107,112)(108,114)(111,123)(117,122)(118,126)(121,127)(124,125);(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)(83,86)(84,87)(88,95)(89,93)(90,92)(91,94)(96,97)(98,106,109,119)(99,103)(100,104)(101,105)(102,108,116,113)(107,122,110,121)(111,117)(112,118)(114,120)(115,124,126,123)(125,127)Polycyclic presentation:
2^8,79|a^2=eh,b^2=h,c^2=d,d^2=g,f^2=h,b^a=bd,c^a=ce,f^a=fg,i^a=i^78,c^b=cf,d^b=dg,e^b=eg,f^b=fh,i^b=i^78,f^c=fgh1522 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 74355);