Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C32×C79 ⋊ C2 |
| Sylow | C79 ⋊ #13615 |
298 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 147 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)(84,87,90,105)(85,92,96,95)(86,98,102,101)(88,103,109,136)(89,104,112,137)(91,113,115,123)(93,116,120,119)(94,117,122,110)(97,124,125,132)(99,126,129,128)(100,127,131,111)(106,133,139,138)(107,134,141,140)(108,135,142,145)(114,121,143,147)(118,130,144,146);(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)(84,86)(85,91)(87,98)(88,107)(89,111)(90,102)(92,113)(93,118)(94,121)(95,123)(96,115)(97,106)(99,108)(100,104)(101,105)(103,134)(109,141)(110,114)(112,127)(116,130)(117,143)(119,146)(120,144)(122,147)(124,133)(125,139)(126,135)(128,145)(129,142)(131,137)(132,138)(136,140);(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)(80,81,82,83)(84,85)(86,97)(87,92)(88,106)(89,110)(90,96)(91,99)(93,108)(94,104)(95,105)(98,124)(100,130)(101,132)(102,125)(103,133)(107,121)(109,139)(111,118)(112,117)(113,126)(114,140)(115,129)(116,135)(119,145)(120,142)(122,137)(123,128)(127,144)(131,146)(134,143)(136,138)(141,147)Polycyclic presentation:
2^8,79|a^2=e,c^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^78,d^b=df,f^b=fg,g^b=gh,i^b=i^785068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90003);