Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7378 |
170 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 39 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:
(1,3,6,17,8,19,7,18)(2,9,12,33,13,34,28,47)(4,15,22,49,23,50,29,52)(5,24,26,51,27,53,30,16)(10,36,20,46,39,14,38,45)(11,40,41,62,43,61,42,32)(21,55,56,59,25,31,35,48)(37,63,58,44,64,57,54,60)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142);(1,2,5,11,7,12,26,42,8,13,27,43,6,28,30,41)(3,14,16,44,18,46,24,57,19,36,51,60,17,45,53,63)(4,20,21,54,22,10,35,37,23,38,25,58,29,39,56,64)(9,31,32,50,33,59,61,52,34,55,62,15,47,48,40,49)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)(130,131)(132,133)(134,135)(136,137)(138,139)(140,141)(142,143)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=h,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^78,c^b=cgh,d^b=de,g^b=gh,i^b=i^78,d^c=dgh,e^c=eh,e^d=eh1516 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57004);