Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Sylow | C97 ⋊ C4×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #13095525 |
| commutator, hypocenter | C97 | C4×C32 |
| Frattini | C4 | C2×#13095525 |
| Fitting | C8×C97 | C16 |
| socle | C2×C97 | C4×C16 |
| prime radical | C2×D97 | C2×C16 |
| cosocle | #15690905 | C22 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C8 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 2 generators on 129 points:
(1,2,15,4)(3,27,93,74)(5,39,29,30)(6,19,44,21)(7,8,56,41)(9,31,78,79)(10,49,80,52)(11,32,94,34)(12,13,59,62)(14,45,72,88)(16,46,89,48)(17,18,40,86)(20,75,97,90)(22,23,60,64)(24,25,50,81)(26,67,36,85)(28,68,82,69)(33,51,37,38)(35,55,91,77)(42,57,83,84)(43,54,61,63)(47,76,73,71)(53,87,96,58)(65,66,70,92);(2,3,5,10,7,12,17,29,6,11,16,28,4,9,14,26)(8,20,22,56,24,58,25,54,23,57,51,89,21,55,59,81)(13,33,35,80,37,50,83,86,36,48,82,39,34,18,47,49)(15,40,42,87,19,52,53,90,43,67,79,91,41,38,32,78)(27,61,63,30,65,70,44,45,64,88,60,71,62,46,66,69)(31,73,75,85,77,72,95,68,76,92,97,84,74,93,96,94)(98,99,100,104,101,105,108,114,102,106,109,115,111,117,120,124,103,107,110,116,112,118,121,125,113,119,122,126,123,127,128,129)Polycyclic presentation:
2^7,97|a^2=c,b^2=d,c^2=e,e^2=f,f^2=g,h^a=h^8,h^b=h^75,h^c=h^64,h^d=h^96,h^e=h^22,h^f=h^96176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21778);