Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #13230 |
436 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(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)(80,83,99,128,86,102,85,101)(81,88,106,127,91,111,90,98)(82,94,119,138,97,122,96,121)(84,100,117,140,93,116,105,131)(87,107,135,142,109,137,108,125)(89,110,133,129,103,132,112,126)(92,113,124,141,115,139,114,118)(95,120,134,143,104,130,123,136);(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)(80,82,93,95,86,97,84,104)(81,87,103,115,91,109,89,92)(83,94,116,120,102,122,100,130)(85,96,117,123,99,119,105,134)(88,107,132,139,111,137,110,113)(90,108,133,124,106,135,112,114)(98,125,129,141,127,142,126,118)(101,121,140,136,128,138,131,143);(80,81)(82,92,97,115)(83,98)(84,103,93,89)(85,106)(86,91)(87,95)(88,101)(90,99)(94,118,122,141)(96,124,119,114)(100,129,116,126)(102,127)(104,109)(105,112,117,133)(107,136)(108,134)(110,131,132,140)(111,128)(113,138,139,121)(120,125)(123,135)(130,142)(137,143)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,c^2=eh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fg,g^b=gh,i^c=i^78,f^d=fh931 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 87908);