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 | C8 | #13095525 |
| commutator, hypocenter | C97 | C128 |
| Fitting | C8×C97 | C16 |
| socle, prime radical | C2×C97 | C64 |
| cosocle | #20261817 | C2 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C8 | |
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:
(2,3,4,8,5,9,12,22,6,10,13,23,15,25,33,52)(7,17,18,46,19,47,21,45,20,48,39,84,49,83,51,68)(11,28,29,66,30,38,69,75,31,67,70,76,72,14,36,37)(16,41,42,87,43,54,88,89,44,32,26,60,40,74,61,94)(24,55,56,86,35,81,82,50,57,85,34,77,78,59,79,80)(27,62,63,73,64,58,92,53,65,90,96,71,97,93,95,91)(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);(1,2,7,21,30,73,70,65,3,11,32,75,22,53,61,94,17,42,25,59,77,43,5,16,45,67,74,8,24,58,55,18,38,13,35,36,83,63,92,87,64,62,82,84,19,50,56,33,76,66,12,34,80,49,46,79,90,97,60,95,96,47,69,78,4,14,39,85,93,57,23,54,28,68,44,15,40,86,81,9,26,48,89,88,91,52,37,41,31,10,27,29,71,72,51,20,6)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^8,h^b=h^64,h^c=h^22,h^d=h^96176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21767);