Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #8495 |
620 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 107 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)(80,82,84,89)(81,86,87,95)(83,91,92,88)(85,93,90,94)(96,99)(97,103)(98,107)(100,113)(101,112)(102,115)(104,117)(105,119)(106,120)(108,125)(109,124)(110,118)(111,114)(116,122)(121,127)(123,126);(80,81)(82,86)(83,90)(84,87)(85,92)(88,93)(89,95)(91,94)(96,98,101,109)(97,102,104,106)(99,107,112,124)(100,108,114,126)(103,115,117,120)(105,116,118,121)(110,127,119,122)(111,123,113,125);(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)(81,85,87,90)(86,94,95,93)(96,97)(98,106,109,102)(99,110,112,119)(100,105)(101,104)(103,111,117,113)(107,122)(108,121,126,116)(114,118)(115,123)(120,125)(124,127)Polycyclic presentation:
2^8,79|a^2=f,b^2=g,c^2=h,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,i^a=i^78,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg,i^c=i^781048 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 61945);