Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #8476 |
678 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 678 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) = 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)(80,82,84,90)(81,86,85,88)(83,92,91,89)(87,95,93,94)(96,99,101,112)(97,103,105,110)(98,107,109,124)(100,115,116,111)(102,117,118,126)(104,121,113,119)(106,122,120,127)(108,125,114,123);(81,85)(83,91)(86,88)(89,92)(96,98)(97,102)(99,107)(100,114)(101,109)(103,117)(104,120)(105,118)(106,113)(108,116)(110,126)(111,125)(112,124)(115,123)(119,122)(121,127);(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)(80,81,83,87,84,85,91,93)(82,88,89,95,90,86,92,94)(96,97)(98,106)(99,110,112,103)(100,113)(101,105)(102,108)(104,116)(107,122,124,127)(109,120)(111,119,115,121)(114,118)(117,123,126,125)Polycyclic presentation:
2^8,79|a^2=f,c^2=g,d^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,i^a=i^78,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fg1756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 61824);