Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×#362×C79 ⋊ C2 |
| Sylow | C79 ⋊ #13006 |
472 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 109 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 — 3 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)(80,83)(81,89)(82,95)(84,107)(85,104)(86,113)(87,105)(88,115)(90,124)(91,121)(92,126)(93,122)(94,128)(96,136)(97,134)(98,137)(99,135)(100,114)(101,109)(102,111)(103,108)(106,119)(110,132)(112,131)(116,139)(117,130)(118,143)(120,127)(123,142)(125,141)(129,140)(133,138);(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)(80,82,85,110,87,99,111,97)(81,88,114,116,93,118,91,125)(83,95,104,132,105,135,102,134)(84,98,108,96,109,138,86,112)(89,115,100,139,122,143,121,141)(90,123,127,129,106,117,92,94)(101,133,113,131,107,137,103,136)(119,130,126,128,124,142,120,140);(80,81,86,92,111,114,109,90,87,93,108,127,85,91,84,106)(82,94,98,88,110,117,96,125,99,129,138,118,97,123,112,116)(83,100,103,119,102,122,107,126,105,121,113,124,104,89,101,120)(95,130,133,139,132,140,131,115,135,142,137,141,134,128,136,143)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,d^2=g,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,i^c=i^781288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 86586);