Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13499 |
276 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 276 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)(80,83,85,107,86,100,104,102)(81,88,91,114,92,113,90,115)(82,93,96,99,97,123,95,125)(84,103,105,101,106,136,120,135)(87,108,110,140,111,137,121,112)(89,116,118,98,119,132,117,131)(94,126,128,124,129,134,127,143)(109,130,138,122,139,133,142,141);(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)(81,87)(83,99)(84,94)(85,96)(86,97)(88,112)(89,109)(90,121)(91,110)(92,111)(93,102)(95,104)(98,130)(100,125)(101,134)(103,124)(105,128)(106,129)(107,123)(108,114)(113,140)(115,137)(116,141)(117,142)(118,138)(119,139)(120,127)(122,132)(126,135)(131,133)(136,143);(80,81,84,89,104,90,120,117,86,92,106,119,85,91,105,118)(82,87,94,109,95,121,127,142,97,111,129,139,96,110,128,138)(83,98,101,114,107,132,136,113,100,131,135,115,102,116,103,88)(93,122,124,140,99,133,134,137,123,141,143,112,125,130,126,108)Polycyclic presentation:
2^8,79|a^2=e,c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eg,f^c=fh,g^c=gh,i^c=i^782224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89463);