Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #11970 |
284 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 127 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)(81,86)(85,88)(87,94)(93,95)(96,99,103,102)(97,105,108,114)(98,110,113,118)(100,115,117,101)(104,119,122,121)(106,107,123,125)(109,120,126,124)(111,116,127,112);(80,82,84,90)(81,85,86,88)(83,91,92,89)(87,95,94,93)(96,98)(97,104)(99,110)(100,116)(101,111)(102,118)(103,113)(105,119)(106,124)(107,109)(108,122)(112,117)(114,121)(115,127)(120,123)(125,126);(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)(80,81,83,87,84,86,92,94)(82,88,89,95,90,85,91,93)(96,97,101,107,102,105,117,123,103,108,115,125,99,114,100,106)(98,109,112,122,110,124,111,119,113,126,116,104,118,120,127,121)Polycyclic presentation:
2^8,79|a^2=e,b^2=f,c^2=h,d^2=gh,e^2=fg,g^2=h,b^a=bd,c^a=cfh,d^a=df,i^a=i^78,d^b=dg,e^b=efg,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, 80994);