Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #11861 |
310 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 310 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)(80,83,84,92)(81,86,87,95)(82,89,90,91)(85,93,88,94)(96,99,102,114,103,115,101,100)(97,105,107,106,108,116,118,125)(98,110,112,111,113,117,119,127)(104,120,122,124,123,126,121,109);(80,82)(81,85)(83,91)(84,90)(86,94)(87,88)(89,92)(93,95)(96,98)(97,104)(99,111)(100,117)(101,112)(102,119)(103,113)(105,124)(106,120)(107,121)(108,123)(109,116)(110,114)(115,127)(118,122)(125,126);(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)(82,88,90,85)(83,86)(84,87)(89,94,91,93)(92,95)(96,97,101,107,103,108,102,118)(98,109,112,126,113,124,119,120)(99,106,100,116,115,125,114,105)(104,117,121,127,123,110,122,111)Polycyclic presentation:
2^8,79|a^2=e,c^2=f,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,i^a=i^78,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,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, 80532);