Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13525 |
294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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)(81,89)(82,95)(84,108)(85,103)(86,113)(87,106)(88,114)(90,124)(91,120)(92,129)(93,122)(94,130)(96,135)(97,131)(98,133)(99,102)(100,134)(101,127)(104,111)(105,109)(107,139)(110,138)(112,132)(115,140)(116,142)(117,119)(118,137)(121,125)(123,143)(126,136)(128,141);(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)(80,82,87,100)(81,88,93,118)(83,102)(84,107,111,97)(85,112,110,96)(86,98,109,99)(89,119)(90,123,127,115)(91,128,126,94)(92,116,125,117)(95,113)(101,130)(103,131)(104,135)(105,134)(106,133)(108,132)(114,129)(120,140)(121,137)(122,142)(124,141)(136,143)(138,139);(80,81,84,90,109,125,85,91,87,93,111,127,86,92,110,126)(82,94,96,116,98,115,97,118,100,128,112,117,99,123,107,88)(83,101,103,89,113,120,108,129,106,124,138,122,105,136,104,121)(95,119,131,130,102,137,135,143,134,142,139,141,133,114,132,140)Polycyclic presentation:
2^8,79|a^2=d,b^2=de,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=egh,g^b=gh,i^b=i^78,d^c=dgh,e^c=egh,f^c=fh,g^c=gh1756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89620);