Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12827 |
376 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 376 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)(80,83,86,101)(81,88,92,115)(82,94,97,125)(84,102,104,99)(85,107,108,136)(87,109,111,141)(89,114,91,113)(90,118,119,131)(93,120,112,139)(95,126,128,123)(96,129,106,138)(98,116,132,105)(100,134,135,103)(110,122,117,143)(121,130,142,140)(124,137,133,127);(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)(80,82,86,97)(81,87)(83,94,101,125)(84,95,104,128)(85,106,108,96)(88,109)(89,93)(90,117)(91,112)(92,111)(98,130)(99,123,102,126)(100,133,135,124)(103,137,134,127)(105,121)(107,138,136,129)(110,119)(113,139)(114,120)(115,141)(116,142)(118,143)(122,131)(132,140);(80,81)(82,93,97,112)(83,98)(84,89)(85,105)(86,92)(87,95,111,128)(88,100)(90,103)(91,104)(94,122,125,143)(96,110,106,117)(99,131)(101,132)(102,118)(107,113)(108,116)(109,138,141,129)(114,136)(115,135)(119,134)(120,133,139,124)(121,137,142,127)(123,140,126,130)Polycyclic presentation:
2^8,79|a^2=f,b^2=dg,c^2=h,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^b=eh,f^b=fh,i^b=i^78,d^c=dh,e^c=eg,g^c=gh,i^c=i^781288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85690);