Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C79×#1426 ⋊ C2 |
| Sylow | C79 ⋊ #13638 |
182 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 79 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)(80,83,86,101)(81,88,92,91)(82,95,97,127)(84,99,105,135)(85,100,107,103)(87,109,110,112)(89,113,116,115)(90,114,117,106)(93,118,120,122)(94,124,125,104)(96,126,130,139)(98,128,132,129)(102,134,136,108)(111,119,138,137)(121,140,142,133)(123,141,143,131);(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)(80,82,94,123,85,98,84,96,101,127,104,131,103,129,135,139,86,97,125,143,107,132,105,130,83,95,124,141,100,128,99,126)(81,87,108,133,90,111,89,93,88,109,102,121,114,119,113,118,92,110,134,140,117,138,116,120,91,112,136,142,106,137,115,122);(80,81)(82,93,97,120)(83,88)(84,102)(85,106)(86,92)(87,96,110,130)(89,104)(90,100)(91,101)(94,113)(95,118,127,122)(98,133,132,140)(99,134)(103,117)(105,136)(107,114)(108,135)(109,126,112,139)(111,123,138,143)(115,125)(116,124)(119,141,137,131)(121,129,142,128)Polycyclic presentation:
2^8,79|a^2=e,b^2=dfg,c^2=h,d^2=fgh,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,e^b=eh,i^c=i^781066 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90087);