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 ⋊ #12893 |
234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 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)(80,83,84,102,107,134,108,135,87,104,95,123,86,105,85,103)(81,89,90,114,109,133,91,101,93,115,117,136,92,116,106,132)(82,96,97,125,129,142,130,143,100,127,122,141,99,128,98,126)(88,110,94,118,119,138,111,131,113,124,121,139,112,137,120,140);(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)(80,82,95,122,107,129,85,98,87,100,84,97,86,99,108,130)(81,88,90,94,109,119,91,111,93,113,117,121,92,112,106,120)(83,96,123,141,134,142,103,126,104,127,102,125,105,128,135,143)(89,110,114,118,133,138,101,131,115,124,136,139,116,137,132,140);(80,81)(82,94)(83,101,104,132)(84,106)(85,90)(86,109)(87,93)(88,97)(89,103,115,135)(91,95)(92,107)(96,124,127,110)(98,119)(99,111)(100,121)(102,133,123,116)(105,136,134,114)(108,117)(112,130)(113,122)(118,126,139,143)(120,129)(125,131,141,140)(128,137,142,138)Polycyclic presentation:
2^8,79|a^2=f,b^2=eg,c^2=d,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,i^b=i^78,f^c=fh,i^c=i^781300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85964);