Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7425 |
136 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 41 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 — 2 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)(80,82,88,100)(81,89,93,119)(83,102,107,130)(84,99,87,96)(85,111,86,113)(90,121,124,142)(91,116,114,118)(92,125,126,143)(94,127,129,101)(95,105,131,106)(97,103,98,109)(104,140,108,138)(110,141,112,117)(115,120,132,139)(122,133,123,134)(128,137,135,136);(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)(80,81,84,91,88,93,87,114)(82,94,96,128,100,129,99,135)(83,101,104,136,107,127,108,137)(85,110,109,126,86,112,103,92)(89,115,116,131,119,132,118,95)(90,120,122,105,124,139,123,106)(97,133,111,142,98,134,113,121)(102,125,138,141,130,143,140,117)Polycyclic presentation:
2^8,79|a^2=d,b^2=h,c^2=fgh,d^2=h,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,i^a=i^78,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,i^b=i^78,d^c=dh1747 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57145);