Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C79 ⋊ #7198 |
192 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 91 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 — 2 generators on 127 points:
(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)(80,82,83,90,84,89,91,95)(81,85,87,92,86,88,93,94)(96,98)(97,103)(99,112)(100,114)(101,109)(102,111)(104,120)(105,122)(106,108)(107,119)(110,116)(113,126)(115,117)(118,124)(121,125)(123,127);(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)(80,81)(82,88,89,85)(83,87)(84,86)(90,94,95,92)(91,93)(96,97,100,105,101,106,115,123,102,107,116,124,99,104,113,121)(98,108,110,125,109,119,126,122,111,120,114,127,112,103,117,118)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=h,d^2=g,e^2=f,g^2=h,b^a=bc,c^a=cf,c^b=ch,d^b=dfh,i^b=i^783196 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56548);