Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7438 |
174 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 37 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:
(1,3,5,25)(2,8,24,31)(4,20,21,47)(6,17,28,15)(7,18,27,16)(9,34,39,61)(10,38,12,33)(11,32,57,30)(13,41,43,63)(14,45,52,56)(19,42,36,62)(22,49,55,48)(23,53,26,50)(29,46,54,51)(35,60,40,59)(37,44,64,58)(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)(3,13,15,42)(4,19)(5,24,27,10)(6,11)(7,12)(8,14,30,47)(9,21,36,26)(16,43,17,34)(18,44,51,61)(20,38,48,32)(22,41)(23,39)(25,58,46,62)(28,57,54,40)(29,35)(31,52,60,53)(33,49,59,50)(37,55,63,45)(56,64)(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=eg,c^2=fg,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^78,c^b=cf,d^b=defgh,e^b=eh,g^b=gh,i^b=i^78,d^c=dg,e^c=eh,e^d=eh1513 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57184);