Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #13625 |
290 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 290 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)(80,83,87,86)(81,89,92,110)(82,94,98,96)(84,99,105,104)(85,100,109,108)(88,112,115,113)(90,116,120,138)(91,106,122,119)(93,124,128,126)(95,111,133,131)(97,107,135,130)(101,136,123,141)(102,137,132,139)(103,117,129,140)(114,125,142,121)(118,134,127,143);(80,82)(81,88)(83,94)(84,102)(85,107)(86,96)(87,98)(89,112)(90,118)(91,121)(92,115)(93,123)(95,129)(97,108)(99,137)(100,135)(101,128)(103,133)(104,139)(105,132)(106,114)(109,130)(110,113)(111,140)(116,134)(117,131)(119,142)(120,127)(122,125)(124,141)(126,136)(138,143);(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)(80,81,83,89,87,92,86,110)(82,93,96,126,98,128,94,124)(84,101,99,136,105,123,104,141)(85,106,100,122,109,119,108,91)(88,111,113,95,115,131,112,133)(90,117,116,129,120,140,138,103)(97,134,130,118,135,143,107,127)(102,121,139,142,132,125,137,114)Polycyclic presentation:
2^8,79|a^2=e,c^2=h,d^2=fgh,e^2=h,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^78,d^b=dfh,e^b=eh,f^b=fg,g^b=gh1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90044);