Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #11497 |
288 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 288 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) = 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)(80,83,84,91)(81,86,87,93)(82,89,90,95)(85,92,88,94)(96,99,100,115,102,116,103,101)(97,105,117,107,108,114,106,124)(98,110,118,127,113,119,111,112)(104,120,109,121,122,125,123,126);(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)(80,82,84,90)(81,85,87,88)(83,89,91,95)(86,92,93,94)(96,98,102,113)(97,104,108,122)(99,110,116,119)(100,118,103,111)(101,112,115,127)(105,120,114,125)(106,123,117,109)(107,121,124,126);(80,81)(82,88,90,85)(83,86)(84,87)(89,94,95,92)(91,93)(96,97,101,107,103,106,116,105,102,108,115,124,100,117,99,114)(98,109,112,125,111,104,119,121,113,123,127,120,118,122,110,126)Polycyclic presentation:
2^8,79|a^2=e,b^2=f,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,d^b=dh,e^b=efh,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, 78917);