Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C79×#1358 ⋊ C2 |
| Sylow | C79 ⋊ #11990 |
268 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 69 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)(81,88,92,113)(82,94)(84,100)(85,101)(86,99)(87,107,110,134)(89,114,91,98)(90,112,118,139)(93,120,109,132)(95,123)(96,124)(97,122)(102,130)(103,111)(104,128)(105,131)(106,129)(108,133,135,143)(115,140,117,127)(116,138,121,142)(119,137)(125,141)(126,136);(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)(80,82,86,97)(81,87,92,110)(83,94,99,122)(84,102,104,125)(85,105,106,96)(88,107,113,134)(89,109,91,93)(90,116,118,121)(95,103,126,119)(98,120,114,132)(100,130,128,141)(101,131,129,124)(108,115,135,117)(111,136,137,123)(112,138,139,142)(127,133,140,143);(80,81,84,90,85,89,103,115,86,92,104,118,106,91,119,117)(82,93,95,121,96,110,125,135,97,109,126,116,105,87,102,108)(83,98,100,127,101,88,111,112,99,114,128,140,129,113,137,139)(94,107,123,133,124,120,141,142,122,134,136,143,131,132,130,138)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,c^2=g,d^2=g,e^2=fg,f^2=h,b^a=bd,c^a=cf,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,d^c=dh,i^c=i^78931 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 81087);