Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13691 |
200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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)(82,94)(84,99)(85,107)(86,111)(87,102)(88,112)(90,114)(91,119)(92,121)(93,117)(95,124)(96,128)(97,125)(98,106)(100,108)(101,105)(103,116)(104,136)(109,141)(110,115)(113,132)(118,143)(120,130)(122,142)(123,127)(126,135)(129,134)(131,138)(133,139)(137,140);(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)(80,82,84,95,87,97,105,86)(81,88,90,92,93,110,103,122)(83,94,99,124,102,125,101,111)(85,96,104,126,109,129,137,108)(89,112,114,121,117,115,116,142)(91,113,118,106,120,138,127,139)(98,130,131,123,133,119,132,143)(100,107,128,136,135,141,134,140);(80,81)(82,92)(83,98)(84,103)(85,106)(86,110)(87,93)(88,95)(89,100)(90,105)(91,108)(94,123)(96,127)(97,122)(99,132)(101,131)(102,133)(104,113)(107,121)(109,139)(111,119)(112,136)(114,134)(115,140)(116,128)(117,135)(118,129)(120,126)(124,130)(125,143)(137,138)(141,142)Polycyclic presentation:
2^8,79|b^2=d,d^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ef,f^a=fg,g^a=gh,i^b=i^78,e^c=ef,f^c=fg,g^c=gh1051 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90380);