Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #7495 |
280 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,3)(2,7,10,25)(4,17)(5,21)(6,16)(8,24,27,12)(9,28,29,32)(11,30,20,26)(13,19)(14,22)(15,23)(18,31)(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)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110);(1,2,4,9,6,10,19,29)(3,12,14,7,16,24,31,25)(5,20,22,27,23,11,18,8)(13,26,15,28,17,30,21,32)(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)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,d^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fg,g^a=gh,i^a=i^78,c^b=ch,d^b=dfh,f^b=fh,g^b=gh,i^b=i^78,d^c=dgh,f^c=fh,f^d=fh1042 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57338);