Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7240 |
148 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(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)(80,82,83,93,98,120,128,133,86,96,101,123,85,95,100,122)(81,87,88,92,90,107,97,118,91,108,111,119,116,137,127,141)(84,94,99,121,129,132,109,134,104,125,102,105,103,124,130,142)(89,112,110,135,114,139,126,143,115,106,113,117,138,136,131,140);(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)(80,81,84,89,85,90,103,114,86,91,104,115,98,116,129,138)(82,92,94,117,95,118,124,140,96,119,125,135,120,141,132,143)(83,97,99,126,100,111,130,113,101,127,102,131,128,88,109,110)(87,105,106,133,107,134,136,93,108,121,112,122,137,142,139,123)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fgh,d^2=f,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,i^a=i^78,d^b=dg,d^c=dh2224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56665);