Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7429 |
196 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 41 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 — 2 generators on 119 points:
(1,2)(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)(81,84,85,86)(82,87)(88,90,92,102,93,103,108,116)(89,94,97,109,98,110,106,115)(91,100,104,101,105,117,107,118)(95,99,96,112,111,114,113,119);(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)(88,89,91,96,93,98,105,113)(90,99,101,110,103,114,118,94)(92,106,107,111,108,97,104,95)(100,115,116,112,117,109,102,119)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^78,c^b=cf,d^b=defgh,i^b=i^78,d^c=dgh,e^d=eh1042 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57157);