Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #13612 |
304 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 67 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)(81,88)(82,94)(84,100)(85,101)(86,99)(87,108)(89,115)(90,113)(91,112)(92,114)(93,120)(95,121)(96,122)(97,98)(102,133)(103,131)(104,128)(105,127)(106,132)(107,129)(109,134)(110,125)(111,135)(116,140)(117,139)(118,138)(119,141)(123,130)(124,126)(136,142)(137,143);(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)(80,82,86,97)(81,87,92,111)(83,98,99,94)(84,102,104,123)(85,105,107,96)(88,108,114,135)(89,110,91,93)(90,117,119,136)(95,103,124,106)(100,130,128,133)(101,122,129,127)(109,118,137,116)(112,120,115,125)(113,139,141,142)(121,132,126,131)(134,138,143,140);(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)(80,81,84,90,86,92,104,119)(82,93,95,117,97,110,124,136)(83,88,100,113,99,114,128,141)(85,91,106,118,107,89,103,116)(87,102,109,105,111,123,137,96)(94,120,121,139,98,125,126,142)(101,112,132,138,129,115,131,140)(108,133,134,127,135,130,143,122)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^78,c^b=cg,d^b=dgh,e^b=efg,f^b=fh,i^b=i^78,d^c=dh,e^d=egh,f^e=fh922 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89990);