Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C79 ⋊ #11416 |
228 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 123 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)(82,88)(85,91)(89,94)(92,95)(97,103)(102,114)(104,110)(105,116)(106,118)(115,124)(117,125)(119,122);(80,82)(81,85)(83,89)(84,88)(86,92)(87,91)(90,94)(93,95)(96,98,99,108,100,109,111,123,101,107,112,120,113,121,126,127)(97,102,104,115,105,106,117,119,103,114,110,124,116,118,125,122);(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)(80,81,83,86,84,87,90,93)(82,85,89,92,88,91,94,95)(96,97)(98,106,107,118)(99,110)(100,105)(101,103)(102,109,114,121)(104,112)(108,122,120,119)(111,125)(113,116)(115,127,124,123)(117,126)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,b^a=bg,c^a=ch,e^a=eh,i^a=i^78,c^b=cf,d^b=dh2248 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 78624);