Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12885 |
234 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 111 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)(80,83,84,102,87,105,86,103)(81,89,90,118,93,120,92,100)(82,95,96,101,99,129,97,131)(85,108,106,136,110,104,109,116)(88,112,94,117,115,127,113,134)(91,123,121,133,125,119,124,139)(98,132,130,137,107,128,111,135)(114,141,126,143,122,140,138,142);(80,82,84,97,87,99,86,96)(81,88,90,113,93,115,92,94)(83,101,103,129,105,131,102,95)(85,107,106,130,110,98,109,111)(89,117,100,127,120,134,118,112)(91,122,121,126,125,114,124,138)(104,137,136,128,108,135,116,132)(119,143,133,140,123,142,139,141);(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)(80,81,85,91,86,92,109,124,87,93,110,125,84,90,106,121)(82,94,98,126,96,115,130,122,99,113,107,138,97,88,111,114)(83,100,104,133,103,120,136,123,105,118,108,139,102,89,116,119)(95,127,128,141,101,134,135,143,129,112,132,140,131,117,137,142)Polycyclic presentation:
2^8,79|a^2=f,b^2=e,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,i^a=i^78,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85935);