Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #12596 |
310 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 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)(84,87,88,89,90,104,105,106)(85,92,93,111,95,94,96,102)(86,98,99,113,101,103,114,100)(91,107,97,112,108,110,109,115);(80,81,82,83)(84,86)(85,91,95,108)(87,103)(88,99)(89,100)(90,101)(92,110,94,107)(93,97,96,109)(98,104)(102,112,111,115)(105,114)(106,113);(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)(84,85,89,94,88,96,104,111,90,95,106,92,105,93,87,102)(86,97,100,112,99,91,98,107,101,109,113,115,114,108,103,110)Polycyclic presentation:
2^8,79|a^2=e,b^2=f,c^2=d,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=dh,f^a=fh,i^a=i^78,c^b=ch,e^c=eh3184 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 84588);