Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #11421 |
522 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 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)(89,96)(95,110)(97,99)(98,112)(100,113)(111,115)(114,116)(117,119);(80,82)(81,84)(83,87)(85,86)(88,90)(89,95)(91,104)(92,106)(93,101)(94,103)(96,110)(97,113)(98,115)(99,100)(102,108)(105,118)(107,109)(111,112)(114,117)(116,119);(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)(82,86,87,84)(83,85)(88,89,92,98,93,99,107,116,94,96,108,112,91,97,105,114)(90,100,102,117,101,110,118,115,103,113,106,119,104,95,109,111)Polycyclic presentation:
2^8,79|a^2=e,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,i^a=i^78,d^b=dh,e^b=efh2236 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 78640);