Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #11894 |
394 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 394 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) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 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)(82,87)(84,86)(88,91,94,109,95,110,93,92)(89,97,99,108,101,111,100,98)(90,103,106,104,107,112,105,119)(96,113,114,102,116,117,115,118);(80,82)(81,84)(83,87)(85,86)(88,90)(89,96)(91,104)(92,112)(93,106)(94,105)(95,107)(97,102)(98,117)(99,115)(100,114)(101,116)(103,109)(108,113)(110,119)(111,118);(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)(82,86,87,84)(83,85)(88,89,93,99,95,101,94,100)(90,102,105,117,107,118,106,113)(91,108,92,111,110,98,109,97)(96,112,114,104,116,103,115,119)Polycyclic presentation:
2^8,79|a^2=e,c^2=g,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,i^a=i^78,c^b=cfh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh1756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 80675);