Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #8527 |
656 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 131 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)(80,83,85,84)(81,87,88,89)(82,91,92,93)(86,94,90,95)(96,99,101,112)(97,103,105,110)(98,107,109,123)(100,111,114,127)(102,115,116,124)(104,117,119,125)(106,120,118,121)(108,122,113,126);(80,82,85,92)(81,86,88,90)(83,91,84,93)(87,94,89,95)(96,98)(97,102)(99,107)(100,113)(101,109)(103,115)(104,118)(105,116)(106,119)(108,114)(110,124)(111,126)(112,123)(117,121)(120,125)(122,127);(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)(80,81,84,89,85,88,83,87)(82,90,91,95,92,86,93,94)(96,97)(98,106,109,118)(99,110,112,103)(100,104)(101,105)(102,108,116,113)(107,121)(111,125,127,117)(114,119)(115,122)(120,123)(124,126)Polycyclic presentation:
2^8,79|a^2=f,b^2=g,c^2=gh,d^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^78,d^b=dh,f^b=fgh,f^c=fh1750 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 62110);