Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #13643 |
164 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 75 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)(81,88,89,92)(82,94)(83,99)(87,105,93,109)(90,115,112,110)(91,111,113,117)(96,119)(97,120)(100,126)(101,108)(106,131,114,129)(107,130,118,134)(116,138,136,135)(123,139)(127,133)(132,143,137,142);(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)(83,98)(84,96)(85,97)(86,94)(88,109)(89,93)(90,114)(91,107)(92,105)(95,99)(100,124)(101,125)(102,123)(103,119)(104,120)(106,112)(108,122)(110,129)(111,134)(113,118)(115,131)(116,137)(117,130)(121,126)(127,141)(128,139)(132,136)(133,140)(135,142)(138,143);(80,81,83,90,84,91,100,116,85,92,101,115,102,117,127,138,86,89,99,112,103,113,126,136,104,88,108,110,128,111,133,135)(82,93,95,114,96,118,121,137,97,109,122,131,123,134,140,143,94,87,98,106,119,107,124,132,120,105,125,129,139,130,141,142)Polycyclic presentation:
2^8,79|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=cg,d^a=dh,c^b=ch,e^b=eh,e^c=eh,i^c=i^781780 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90108);