Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7413 |
178 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 43 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 — 2 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)(81,84)(82,87)(85,86)(88,90)(89,94,98,113)(91,104)(92,108)(93,103)(95,99,114,111)(96,115,116,118)(97,117,107,112)(100,106)(101,109)(102,110)(105,119);(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)(82,84)(83,86)(85,87)(88,89,91,96,93,98,106,116)(90,99,101,94,103,111,119,113)(92,107,109,114,110,97,105,95)(100,117,102,115,104,112,108,118)Polycyclic presentation:
2^8,79|a^2=d,b^2=g,c^2=fh,d^2=h,e^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,i^a=i^78,c^b=cf,d^b=def,e^b=eh,d^c=dgh,e^d=eh1276 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57108);