Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C79×#1515 ⋊ C2 |
| Sylow | C79 ⋊ #8502 |
672 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 103 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)(80,82)(81,86)(83,92)(84,91)(85,88)(87,94)(89,95)(90,93)(96,99,102,114)(97,104,107,120)(98,109,111,125)(100,115,101,113)(103,116,117,127)(105,121,106,119)(108,122,118,123)(110,126,112,124);(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)(83,90)(84,91)(86,88)(87,89)(92,93)(94,95)(96,98)(97,103)(99,109)(100,112)(101,110)(102,111)(104,116)(105,118)(106,108)(107,117)(113,126)(114,125)(115,124)(119,122)(120,127)(121,123);(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)(82,89,91,85)(83,88)(84,87)(86,90,94,92)(93,95)(96,97,101,106,102,107,100,105)(98,108,112,103,111,118,110,117)(99,104,113,119,114,120,115,121)(109,123,124,127,125,122,126,116)Polycyclic presentation:
2^8,79|a^2=f,c^2=h,d^2=h,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,i^a=i^78,d^b=dh,e^b=eh,f^b=fgh,i^b=i^78,d^c=dh,e^c=eg,f^c=fg1048 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 61990);