Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #13709 |
154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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)(82,94)(84,99)(85,107)(86,111)(87,102)(88,112)(90,115)(91,121)(92,123)(93,118)(95,126)(96,130)(97,128)(98,132)(100,120)(101,105)(103,117)(104,139)(106,137)(108,114)(109,127)(110,116)(113,133)(119,125)(122,135)(124,134)(129,143)(131,138)(136,141)(140,142);(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)(80,82,84,95,87,97,105,86)(81,88,90,92,93,110,103,124)(83,94,99,126,102,128,101,111)(85,96,104,120,109,131,140,108)(89,112,115,123,118,116,117,134)(91,113,119,106,122,141,129,132)(98,121,133,125,137,135,136,143)(100,127,138,142,114,107,130,139);(80,81,87,93)(82,92,97,124)(83,98)(84,103,105,90)(85,106,109,132)(86,110,95,88)(89,114)(91,120,122,108)(94,125)(96,129,131,119)(99,136)(100,118)(101,133)(102,137)(104,113,140,141)(107,123)(111,135)(112,139)(115,138)(116,142)(117,130)(121,126)(127,134)(128,143)Polycyclic presentation:
2^8,79|a^2=dg,b^2=d,d^2=h,e^2=fgh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ef,f^a=fg,g^a=gh,i^b=i^78,e^c=efh,f^c=fg,g^c=gh1051 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90482);