Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12878 |
362 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 119 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)(80,83,86,110,87,99,85,101)(81,89,109,116,90,117,92,115)(82,94,96,125,97,123,108,126)(84,104,106,100,107,138,105,132)(88,111,113,140,93,121,114,142)(91,120,134,131,118,130,102,98)(95,128,129,124,103,136,137,133)(112,141,127,122,119,143,135,139);(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)(81,88)(83,94)(84,103)(85,108)(86,96)(87,97)(89,111)(90,93)(91,119)(92,114)(95,107)(98,122)(99,123)(100,133)(101,126)(102,127)(104,136)(105,129)(106,137)(109,113)(110,125)(112,118)(115,142)(116,140)(117,121)(120,143)(124,132)(128,138)(130,141)(131,139)(134,135);(80,81,85,92,87,90,86,109)(82,93,96,114,97,88,108,113)(83,98,101,130,99,131,110,120)(84,102,105,118,107,134,106,91)(89,104,115,132,117,138,116,100)(94,122,125,143,123,139,126,141)(95,127,129,119,103,135,137,112)(111,128,140,124,121,136,142,133)Polycyclic presentation:
2^8,79|a^2=f,c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh,i^c=i^781294 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85908);