Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C79 ⋊ #7147 |
188 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(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)(80,82,84,92,85,91,83,90)(81,86,87,89,88,94,93,95)(96,98)(97,102,104,106)(99,108)(100,109)(101,107)(103,114,116,119)(105,115,118,120)(110,122)(111,113)(112,121)(117,125,126,127)(123,124);(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)(82,89,91,95)(83,87)(84,93)(85,88)(86,90,94,92)(96,97,99,103,100,105,110,117,101,104,111,116,112,118,123,126)(98,106,108,119,109,120,122,127,107,102,113,114,121,115,124,125)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,g^2=h,b^a=bc,e^a=efh,i^a=i^78,c^b=ch,d^b=df,i^b=i^782248 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56422);