Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C79×D4⋊C16 ⋊ C2 |
| Sylow | C79 ⋊ #12399 |
290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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)(81,87)(83,84)(86,94)(88,89)(90,95)(91,92)(96,99,102,101)(97,104,106,116)(98,108,110,117)(100,111,115,114)(103,119,121,120)(105,112,123,124)(107,125,126,122)(109,113,118,127);(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)(80,82,85,93)(81,86,89,95)(83,92,84,91)(87,94,88,90)(96,98,102,110)(97,103,106,121)(99,108,101,117)(100,113,115,127)(104,119,116,120)(105,122,123,125)(107,124,126,112)(109,111,118,114);(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)(80,81,85,89)(82,90,93,94)(83,88,84,87)(86,92,95,91)(96,97,102,106)(98,107,110,126)(99,104,101,116)(100,112,115,124)(103,118,121,109)(105,111,123,114)(108,125,117,122)(113,119,127,120)Polycyclic presentation:
2^8,79|a^2=gh,b^2=gh,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,i^a=i^78,d^b=df,e^b=eg,f^b=fh,i^b=i^78,d^c=dg,e^c=eg2224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 83567);