Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| Sylow | C79 ⋊ #8158 |
400 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 147 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)(81,86)(85,91)(87,93)(92,95)(96,99,100,110,101,111,112,123)(97,103,104,116,105,117,118,126)(98,107,109,122,108,121,113,124)(102,114,115,125,106,119,120,127);(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)(80,82,83,88,84,89,90,94)(81,85,87,92,86,91,93,95)(96,98)(97,102,105,106)(99,107)(100,109)(101,108)(103,114,117,119)(104,115,118,120)(110,122)(111,121)(112,113)(116,125,126,127)(123,124);(80,81)(82,85)(83,87)(84,86)(88,92)(89,91)(90,93)(94,95)(96,97,100,104,101,105,112,118)(98,106,109,120,108,102,113,115)(99,103,110,116,111,117,123,126)(107,119,122,127,121,114,124,125)Polycyclic presentation:
2^8,79|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,b^a=bd,c^a=ch,f^a=fg,d^b=dg,i^b=i^782248 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 60104);