Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12296 |
278 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 59 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)(80,83,84,90)(81,86,87,93)(82,88,89,95)(85,91,92,94)(96,98)(97,104)(99,108)(100,114)(101,107)(102,109)(103,116)(105,120)(106,121)(110,123)(111,125)(112,126)(113,124)(115,119)(117,127)(118,122);(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)(80,82)(81,85)(83,88)(84,89)(86,91)(87,92)(90,95)(93,94)(96,97,102,106)(98,104,109,121)(99,111,113,122)(100,105,101,115)(103,112,117,110)(107,119,114,120)(108,125,124,118)(116,126,127,123);(80,81,84,87)(82,85,89,92)(83,86,90,93)(88,94,95,91)(97,103)(98,107)(99,110)(100,101)(104,118)(105,122)(106,117)(108,124)(109,114)(111,115)(112,113)(116,119)(120,127)(121,125)Polycyclic presentation:
2^8,79|a^2=g,b^2=h,c^2=g,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,d^b=df,e^b=eg,f^b=fh,i^b=i^78,d^c=dg,e^c=eh1045 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 82878);