Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #13146 |
312 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 312 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:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,83,84,100,85,102,106,135,87,104,108,101,86,103,107,115)(81,89,90,116,109,133,137,134,92,118,111,99,91,117,105,136)(82,94,95,124,96,126,110,138,98,128,130,125,97,127,129,140)(88,112,93,119,120,141,131,143,114,139,122,132,113,123,121,142);(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)(80,82,84,95,85,96,106,110,87,98,108,130,86,97,107,129)(81,88,111,122,109,120,105,121,92,114,90,93,91,113,137,131)(83,94,100,124,102,126,135,138,104,128,101,125,103,127,115,140)(89,112,99,132,133,141,136,142,118,139,116,119,117,123,134,143);(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)(80,81,85,91,87,92,86,109)(82,93,96,121,98,122,97,131)(83,99,102,134,104,116,103,136)(84,105,106,111,108,137,107,90)(88,110,113,130,114,129,120,95)(89,115,117,100,118,135,133,101)(94,123,126,139,128,141,127,112)(119,140,142,124,132,138,143,125)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,i^a=i^78,i^b=i^78,e^c=eh1300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 87479);