Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9985 |
502 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 502 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) = 5056 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2)(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)(80,83)(81,86)(82,88)(84,90)(85,91)(87,94)(89,95)(92,93)(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);(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)(80,82,84,89)(81,85,87,92)(83,88,90,95)(86,93,94,91)(96,98)(97,103)(99,113)(100,112)(101,110)(102,111)(104,120)(105,119)(106,108)(107,118)(109,114)(115,126)(116,125)(117,121)(122,124)(123,127);(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)(82,85)(83,86)(84,87)(88,91)(89,92)(90,94)(93,95)(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=h,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,i^a=i^78,c^b=cf,d^b=dg,e^b=eg,f^b=fg,i^b=i^78,d^c=dh,f^c=fg,i^c=i^781042 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 70232);