Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C97 ⋊ C2×C64 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #13095525 |
| commutator, hypocenter | C97 | C2×C64 |
| Frattini | C4 | C2×#13095525 |
| Fitting | C8×C97 | C16 |
| socle | C2×C97 | C2×C32 |
| prime radical | C2×D97 | C32 |
| cosocle | #15690900 | C22 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C8 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 2 generators on 161 points:
(1,2)(3,24)(4,7)(5,34)(6,35)(8,11)(9,53)(10,54)(12,15)(13,16)(14,76)(17,89)(18,21)(19,86)(20,47)(22,26)(23,27)(25,39)(28,74)(29,32)(30,56)(31,51)(33,36)(37,78)(38,40)(41,75)(42,45)(43,46)(44,80)(48,57)(49,50)(52,59)(55,92)(58,64)(60,95)(61,63)(62,83)(65,94)(66,68)(69,72)(70,73)(71,84)(77,81)(79,90)(82,85)(87,97)(88,96)(91,93);(2,3,5,9,6,10,14,25,4,8,12,22,13,23,33,52)(7,17,19,47,20,48,21,45,18,46,40,85,49,84,51,68)(11,28,30,67,31,38,71,75,29,66,69,76,70,15,37,39)(16,41,43,87,44,53,88,89,42,32,27,64,35,74,65,96)(24,55,57,73,58,63,93,59,56,90,91,72,92,95,97,94)(26,60,62,80,34,77,79,50,61,86,36,81,78,54,82,83)(98,99,100,105,101,106,110,120,102,107,111,121,114,124,130,140,103,108,112,122,115,125,131,141,117,127,133,143,136,146,150,155,104,109,113,123,116,126,132,142,118,128,134,144,137,147,151,156,119,129,135,145,138,148,152,157,139,149,153,158,154,159,160,161)Polycyclic presentation:
2^7,97|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^8,h^b=h^96,h^c=h^64,h^d=h^22,h^e=h^96176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21792);