Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9416 |
380 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 91 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)(80,83)(81,87)(82,91)(84,90)(85,86)(88,93)(89,92)(94,95)(96,99,101,111)(97,103,104,118)(98,106,107,121)(100,113,114,110)(102,115,117,126)(105,119,112,109)(108,125,123,122)(116,127,124,120);(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)(80,82,84,87,85,93,95,92)(81,86,88,94,89,83,91,90)(96,98)(97,102,104,117)(99,106)(100,108)(101,107)(103,115,118,126)(105,116,112,124)(109,120,119,127)(110,122)(111,121)(113,125)(114,123);(80,81)(82,90,93,94)(83,87,86,92)(84,88)(85,89)(91,95)(96,97)(98,102,107,117)(99,109)(100,112)(101,104)(103,110)(105,114)(106,120,121,127)(108,124,123,116)(111,119)(113,118)(115,125,126,122)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,i^b=i^78,d^c=dgh,e^c=eg,f^c=fgh1048 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 67048);