Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C79 ⋊ #7197 |
184 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 83 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 — 2 generators on 127 points:
(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)(80,82,83,90,84,89,91,95)(81,85,87,92,86,88,93,94)(96,98)(97,102,104,106)(99,108)(100,109)(101,107)(103,114,116,119)(105,115,118,120)(110,122)(111,113)(112,121)(117,125,126,127)(123,124);(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)(80,81)(82,88,89,85)(83,87)(84,86)(90,94,95,92)(91,93)(96,97,99,103,100,105,110,117,101,104,111,116,112,118,123,126)(98,106,108,119,109,120,122,127,107,102,113,114,121,115,124,125)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,d^2=g,e^2=f,g^2=h,b^a=bc,c^a=cf,e^a=eh,c^b=ch,d^b=df,i^b=i^782236 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56546);