Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #11946 |
302 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 127 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)(80,83,85,84)(81,87,88,89)(82,91,92,93)(86,94,90,95)(97,103)(102,117)(104,105)(106,119)(107,125)(118,124)(120,122)(121,123);(80,82)(81,86)(83,91)(84,93)(85,92)(87,94)(88,90)(89,95)(96,98)(97,102)(99,112)(100,110)(101,116)(103,117)(104,120)(105,122)(106,124)(107,121)(108,115)(109,126)(111,113)(114,127)(118,119)(123,125);(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)(80,81,84,89,85,88,83,87)(82,90,91,95,92,86,93,94)(96,97,99,105,113,119,126,121,100,103,114,104,101,106,115,123)(98,107,109,124,116,122,112,117,110,125,108,118,111,120,127,102)Polycyclic presentation:
2^8,79|a^2=e,c^2=fh,d^2=gh,e^2=fg,g^2=h,b^a=bd,c^a=ch,d^a=df,i^a=i^78,d^b=dg,e^b=efg,g^b=gh1762 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 80888);