Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #10648 |
236 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 67 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)(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)(81,87)(82,90)(83,84)(86,93)(88,89)(91,92)(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)(81,86)(83,91)(84,90)(85,92)(87,95)(88,94)(89,93)(96,98,101,109)(97,102,105,116)(99,110,112,107)(100,113,114,108)(103,115,118,126)(104,119,120,106)(111,123,125,124)(117,122,127,121);(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)(80,81,84,87,85,89,83,88)(82,86,90,95,92,93,91,94)(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=g,c^2=d,d^2=g,e^2=h,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,e^c=eh,f^c=fgh,i^c=i^781279 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 74473);