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 | C4 | #15690911 |
| commutator, hypocenter | C97 | C2×C64 |
| Frattini | C2 | C2×#15690911 |
| Fitting | C4×C97 | C32 |
| socle | C2×C97 | C2×C32 |
| prime radical | C2×D97 | C32 |
| cosocle | #15690909 | C22 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C4 | |
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)(3,32)(4,8)(5,43)(6,12)(7,44)(9,13)(10,39)(11,76)(14,19)(15,21)(16,22)(17,78)(18,74)(20,80)(23,35)(24,28)(25,97)(26,29)(27,68)(30,36)(31,37)(33,60)(34,73)(38,57)(40,63)(41,77)(42,46)(45,55)(47,87)(48,52)(49,88)(50,84)(51,72)(53,61)(54,56)(58,95)(59,66)(62,65)(64,91)(67,70)(69,82)(71,86)(75,79)(81,90)(83,85)(89,96)(92,94);(2,3,5,10,6,11,17,33,7,12,18,34,20,35,45,70,4,9,14,29,15,30,32,73,16,31,42,74,36,75,77,76)(8,23,25,63,26,64,68,65,27,39,69,83,28,71,61,43,24,62,66,82,56,94,79,87,67,13,38,40,72,41,81,84)(19,47,49,44,50,88,93,86,51,89,54,22,57,58,95,59,48,85,90,60,91,78,80,96,92,97,21,53,37,52,55,46)(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^28,h^b=h^96,h^c=h^8,h^d=h^64,h^e=h^22,h^f=h^96140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21814);