Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13653 |
188 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 20224 |
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)(80,82,83,81)(85,92)(91,113)(93,116)(94,117)(95,111)(96,123)(101,122)(107,141)(114,143)(115,121)(118,144)(119,132)(120,145)(124,131)(128,147)(133,139);(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)(84,86,90,100)(85,91,92,113)(87,102,106,136)(88,108,110,142)(89,112,109,99)(93,118,116,144)(94,120,117,145)(95,121,111,115)(96,122,123,101)(97,125,127,104)(98,129,130,103)(105,140,138,135)(107,114,141,143)(119,147,132,128)(124,133,131,139)(126,137,134,146);(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)(80,81,83,82)(84,85)(86,96)(87,101)(88,107)(89,111)(90,92)(91,97)(93,104)(94,103)(95,109)(98,128)(99,131)(100,123)(102,120)(105,139)(106,122)(108,118)(110,141)(112,124)(113,127)(114,135)(115,134)(116,125)(117,129)(119,146)(121,126)(130,147)(132,137)(133,138)(136,145)(140,143)(142,144)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,c^2=e,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,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^783172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90167);