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 | C16 | #11715906 |
| commutator, hypocenter | C97 | C4×C32 |
| Frattini | C8 | C2×#11715906 |
| Fitting | C16×C97 | C8 |
| socle | C2×C97 | C4×C16 |
| prime radical | C2×D97 | C2×C16 |
| cosocle | #15690883 | C22 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C16 | |
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:
(1,2,12,4)(3,19,65,44)(5,7,34,28)(6,13,30,15)(8,21,68,40)(9,11,48,37)(10,20,58,23)(14,17,35,54)(16,18,25,49)(22,38,47,76)(24,26,36,75)(27,57,60,88)(29,32,41,43)(31,53,94,59)(33,50,83,61)(39,69,87,70)(42,66,67,55)(45,46,91,89)(51,80,63,85)(52,56,95,81)(62,64,84,74)(71,72,82,93)(73,77,92,96)(78,79,90,86);(2,3,5,9,6,10,4,8)(7,14,16,18,17,26,15,33)(11,22,24,38,25,37,23,19)(12,27,13,31,29,52,28,51)(20,39,41,44,42,21,40,45)(30,54,35,50,55,78,43,67)(32,57,58,72,49,77,34,59)(36,60,62,70,63,46,61,71)(47,73,74,82,75,81,48,53)(56,84,86,76,83,65,85,69)(64,68,87,66,88,79,80,89)(90,94,95,92,93,97,91,96)(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^64,h^b=h^75,h^c=h^22,h^d=h^96,h^e=h^96320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21594);