Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #7401 |
234 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 39 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)(80,82,86,97)(81,87,91,113)(83,99,103,109)(84,104,100,96)(85,95,108,94)(88,115,119,140)(89,120,116,112)(90,111,106,110)(92,117,124,118)(93,126,129,101)(98,133,135,143)(102,137,139,136)(105,130,107,132)(114,128,138,127)(121,125,122,141)(123,142,131,134);(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)(80,81,84,89,86,91,100,116)(82,92,94,123,97,124,95,131)(83,98,101,134,103,135,126,142)(85,106,107,121,108,90,105,122)(87,109,110,137,113,99,111,136)(88,114,117,102,119,138,118,139)(93,125,127,120,129,141,128,112)(96,115,130,133,104,140,132,143)Polycyclic presentation:
2^8,79|a^2=d,b^2=h,c^2=fh,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^78,c^b=cf,d^b=def,f^b=fh,d^c=dg,e^d=eh1039 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57072);