Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #13520 |
238 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 91 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)(80,83,85,105,86,98,102,100)(81,88,91,116,90,110,113,111)(82,92,95,124,94,97,121,119)(84,101,103,99,104,130,131,128)(87,106,108,139,109,133,136,134)(89,112,115,96,114,132,140,125)(93,120,123,118,122,129,141,127)(107,135,137,117,138,142,143,126);(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)(83,97,98,92)(84,93)(85,95)(86,94)(88,106,110,133)(89,107)(90,109)(91,108)(96,117,125,126)(99,127,128,118)(100,124,105,119)(101,129,130,120)(102,121)(103,123)(104,122)(111,134,116,139)(112,135,132,142)(113,136)(114,138)(115,137)(131,141)(140,143);(80,81,84,89,102,113,131,140,86,90,104,114,85,91,103,115)(82,87,93,107,121,136,141,143,94,109,122,138,95,108,123,137)(83,96,99,116,105,132,130,110,98,125,128,111,100,112,101,88)(92,117,118,139,124,142,129,133,97,126,127,134,119,135,120,106)Polycyclic presentation:
2^8,79|a^2=e,b^2=d,c^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eg,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, 89586);