Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13468 |
304 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 89 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)(81,89)(82,96)(84,106)(85,105)(86,113)(87,108)(88,115)(90,123)(91,120)(92,102)(93,126)(94,122)(95,128)(97,103)(98,132)(99,134)(100,136)(101,131)(104,109)(107,111)(110,139)(112,121)(114,129)(116,138)(117,143)(118,142)(119,124)(125,137)(127,130)(133,141)(135,140);(80,82,84,97,86,100,109,114,87,101,110,98,111,127,85,99)(81,88,90,95,93,117,92,116,94,118,91,124,112,133,125,135)(83,103,104,131,107,134,106,136,108,132,105,96,113,129,139,130)(89,119,102,115,121,138,123,141,122,128,137,142,126,140,120,143);(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)(80,81,86,112,87,94,111,93)(82,95,127,116,101,124,100,135)(83,102,113,123,108,137,107,120)(84,92,109,90,110,125,85,91)(88,114,117,97,118,99,133,98)(89,106,121,104,122,139,126,105)(96,115,130,143,131,142,136,141)(103,138,134,119,132,140,129,128)Polycyclic presentation:
2^8,79|a^2=de,b^2=d,d^2=g,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gh,i^a=i^78,c^b=cf,e^b=eh,d^c=dg,e^c=egh,f^c=fg,g^c=gh1288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89277);