Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C79×#2110 ⋊ C2 |
| Sylow | C79 ⋊ #13216 |
434 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 111 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)(81,89,92,117)(82,95,98,127)(84,101)(85,102)(86,104)(87,103)(88,111)(90,116,118,137)(91,99,109,132)(93,119)(94,124)(96,126,129,143)(97,100,108,136)(105,131,110,139)(106,134)(107,138)(112,140)(113,115)(114,130)(120,125)(121,141)(122,133)(123,142,128,135);(80,82,94,123,86,97,84,96,87,98,106,128,85,108,107,129)(81,88,110,120,91,113,90,93,92,114,105,121,109,112,118,122)(83,100,134,143,104,127,138,142,103,136,124,126,102,95,101,135)(89,115,131,133,99,130,137,125,117,140,139,119,132,111,116,141);(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)(80,81)(82,93)(83,99,103,132)(84,105)(85,91)(86,109)(87,92)(88,96)(89,102,117,104)(90,106)(94,118)(95,125,127,141)(97,120)(98,122)(100,133,136,119)(101,137,138,116)(107,110)(108,121)(111,135,130,142)(112,128)(113,123)(114,129)(115,126,140,143)(124,139,134,131)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfgh,c^2=eg,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^78,c^b=cg,e^b=eh,f^c=fh1750 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 87845);