Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #13101 |
630 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(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)(80,83,87,105)(81,89,93,118)(82,96,100,131)(84,108,109,102)(85,104,86,103)(88,113,115,141)(90,119,106,116)(91,117,92,101)(94,122,125,139)(95,126,110,137)(97,132,107,128)(98,130,99,129)(111,138,133,143)(112,135,121,136)(114,134,120,140)(123,142,124,127);(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)(80,82,95,111)(81,88,112,124)(83,96,126,138)(84,107,86,98)(85,99,109,97)(87,100,110,133)(89,113,135,127)(90,94,92,114)(91,120,106,125)(93,115,121,123)(101,134,119,122)(102,132,104,129)(103,130,108,128)(105,131,137,143)(116,139,117,140)(118,141,136,142);(80,81,86,92,87,93,85,91)(82,94,99,124,100,125,98,123)(83,101,104,89,105,117,103,118)(84,106,95,121,109,90,110,112)(88,111,114,107,115,133,120,97)(96,127,130,122,131,142,129,139)(102,136,126,116,108,135,137,119)(113,132,140,143,141,128,134,138)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,i^b=i^78,d^c=dh,e^c=eh,f^c=fh,f^d=fh1165 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 87160);