Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #8327 |
684 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 684 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) = 127 |
| transitive degree | μt(G) = 5056 |
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)(82,88)(85,91)(89,94)(92,95)(96,99,101,112)(97,103,105,110)(98,107,109,123)(100,111,115,127)(102,116,117,124)(104,118,113,125)(106,120,119,121)(108,122,114,126);(80,82)(81,85)(83,89)(84,88)(86,92)(87,91)(90,94)(93,95)(96,98,101,109)(97,102,105,117)(99,107,112,123)(100,114,115,108)(103,116,110,124)(104,119,113,106)(111,126,127,122)(118,121,125,120);(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)(80,81,83,86,84,87,90,93)(82,85,89,92,88,91,94,95)(96,97)(98,106)(99,110,112,103)(100,113)(101,105)(102,108)(104,115)(107,121,123,120)(109,119)(111,118,127,125)(114,117)(116,122,124,126)Polycyclic presentation:
2^8,79|a^2=f,b^2=g,c^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,i^a=i^78,c^b=ch,d^b=dg,f^c=fg1762 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 61006);