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 | C16 | #11715906 |
| commutator, hypocenter | C97 | C2×C64 |
| Frattini | C8 | C2×#11715906 |
| Fitting | C16×C97 | C8 |
| socle | C2×C97 | C2×C32 |
| prime radical | C2×D97 | C32 |
| cosocle | #15690860 | C22 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C16 | |
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)(3,17)(4,6)(5,23)(7,9)(8,32)(10,11)(12,45)(13,15)(14,53)(16,18)(19,66)(20,22)(21,70)(24,51)(25,27)(26,47)(28,29)(30,31)(33,64)(34,36)(35,84)(37,39)(38,52)(40,42)(43,44)(46,76)(48,50)(49,74)(54,75)(55,63)(56,58)(57,68)(59,60)(61,83)(62,93)(65,73)(67,97)(69,87)(71,80)(72,78)(77,86)(79,94)(81,88)(82,85)(89,90)(91,92)(95,96);(2,3,5,8,4,7,10,16)(6,12,14,15,13,22,28,30)(9,19,21,41,20,40,43,32)(11,24,26,49,25,48,23,46)(17,33,35,18,34,38,59,37)(27,51,52,63,44,71,53,81)(29,45,54,72,39,65,47,75)(31,55,57,82,56,36,61,62)(42,67,69,80,68,91,70,74)(50,77,79,87,78,66,83,60)(58,84,86,64,85,73,76,89)(88,92,96,93,95,90,97,94)(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^64,h^b=h^96,h^c=h^22,h^d=h^96320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21613);