Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #7475 |
170 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 170 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) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 2 generators on 143 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)(80,82,84,97,105,128,135,142,108,129,136,141,109,133,103,125,87,96,101,126,85,98,102,127,86,99,106,131,100,124,83,95)(81,88,90,112,117,140,116,123,110,138,137,143,107,130,89,94,93,111,120,139,91,113,118,134,92,114,119,132,115,121,104,122);(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)(82,94,96,122)(83,90)(84,104)(85,107)(86,110)(87,93)(88,95,111,125)(89,101)(91,109)(92,108)(97,130,126,121)(98,132,128,143)(99,134,129,123)(100,117)(102,137)(103,120)(105,115)(106,116)(112,124,139,133)(113,141,140,131)(114,142,138,127)(118,136)(119,135)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fgh,e^2=fh,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,d^b=dh,i^b=i^781768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57278);