Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C128 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C97 ⋊ C128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C16 | #11715906 |
| commutator, hypocenter | C97 | C128 |
| Fitting | C16×C97 | C8 |
| socle, prime radical | C2×C97 | C64 |
| cosocle | #20261574 | C2 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C16 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 225 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 2 generators on 225 points:
(2,3,4,7,5,8,10,16)(6,12,13,15,14,22,28,30)(9,19,20,40,21,41,43,32)(11,24,25,48,26,49,23,46)(17,33,34,59,35,18,37,38)(27,51,52,65,44,71,53,81)(29,45,54,72,36,61,47,75)(31,55,56,82,57,39,63,64)(42,67,68,80,69,91,70,74)(50,77,78,89,79,66,83,62)(58,84,85,60,86,73,76,87)(88,95,94,90,92,96,93,97)(98,99,100,106,101,107,112,127,102,108,113,128,117,132,142,162,103,109,114,129,118,133,143,163,121,136,146,166,152,172,182,197,104,110,115,130,119,134,144,164,122,137,147,167,153,173,183,198,124,139,149,169,155,175,185,200,158,178,188,203,192,207,212,218,105,111,116,131,120,135,145,165,123,138,148,168,154,174,184,199,125,140,150,170,156,176,186,201,159,179,189,204,193,208,213,219,126,141,151,171,157,177,187,202,160,180,190,205,194,209,214,220,161,181,191,206,195,210,215,221,196,211,216,222,217,223,224,225);(1,2,6,15,31,58,82,93,78,63,69,41,66,85,46,74,71,24,50,72,54,25,4,11,27,43,70,89,59,84,61,12,21,8,18,39,65,90,96,67,92,87,37,52,13,29,35,16,32,40,7,17,36,28,53,34,60,88,91,95,97,81,55,33,3,9,22,45,73,38,62,68,20,44,26,10,23,47,75,79,49,51,80,48,76,77,19,42,56,83,94,64,86,57,30,14,5)Polycyclic presentation:
2^7,97|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^64,h^b=h^22,h^c=h^96320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21508);