Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C79×#2178 ⋊ C2 |
| Sylow | C79 ⋊ #13655 |
186 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 67 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)(80,83,85,101,87,102,108,86)(81,89,91,116,93,117,92,109)(82,95,97,126,99,127,98,110)(84,100,105,131,107,132,139,106)(88,112,113,140,115,141,122,114)(90,103,119,133,121,135,120,134)(94,123,124,142,125,143,130,118)(96,104,111,136,129,138,128,137);(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)(81,88)(83,95)(84,104)(85,97)(86,110)(87,99)(89,112)(90,118)(91,113)(92,122)(93,115)(94,103)(96,106)(98,108)(100,111)(101,126)(102,127)(105,136)(107,138)(109,114)(116,140)(117,141)(119,123)(120,143)(121,142)(124,133)(125,135)(128,132)(129,131)(130,134)(137,139);(80,81,85,91,87,93,108,92)(82,94,97,124,99,125,98,130)(83,89,101,116,102,117,86,109)(84,103,105,133,107,135,139,134)(88,111,113,129,115,128,122,96)(90,100,119,131,121,132,120,106)(95,123,126,142,127,143,110,118)(104,112,136,140,138,141,137,114)Polycyclic presentation:
2^8,79|a^2=e,c^2=e,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^b=i^781792 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90174);