Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #11914 |
272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,83,86,99)(81,88)(82,94,97,125)(84,103,104,100)(85,101,107,131)(87,109)(89,117)(90,120)(91,98)(92,115)(93,123)(95,128,129,126)(96,127,106,134)(102,136,108,132)(105,113,116,133)(110,141)(111,138)(112,140)(114,122)(118,130)(119,143)(121,137)(124,142)(135,139);(80,82,86,97)(81,87,92,112)(83,94,99,125)(84,102,104,108)(85,106,107,96)(88,109,115,140)(89,111,91,93)(90,119,122,124)(95,105,129,116)(98,123,117,138)(100,132,103,136)(101,134,131,127)(110,118,135,121)(113,126,133,128)(114,142,120,143)(130,139,137,141);(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)(80,81,84,90,85,91,105,121,86,92,104,122,107,89,116,118)(82,93,95,124,96,87,108,110,97,111,129,119,106,112,102,135)(83,98,100,130,101,115,133,120,99,117,103,137,131,88,113,114)(94,109,126,139,127,138,136,142,125,140,128,141,134,123,132,143)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,c^2=gh,d^2=g,e^2=f,f^2=h,b^a=bd,c^a=cf,d^a=df,g^a=gh,i^a=i^78,d^b=dg,e^b=efgh,f^b=fh,d^c=dh,e^c=eh,e^d=eh1282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 80758);