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 | C4 | #15690911 |
| commutator, hypocenter | C97 | C128 |
| Fitting | C4×C97 | C32 |
| socle, prime radical | C2×C97 | C64 |
| cosocle | #20261859 | C2 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C4 | |
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:
(2,3,4,9,5,10,14,29,6,11,15,30,18,32,42,70,7,12,16,31,19,33,37,73,21,35,43,74,34,75,78,77)(8,23,24,62,25,63,66,65,26,41,67,84,28,71,61,45,27,64,68,82,55,94,76,87,69,13,38,39,72,40,81,83)(17,46,47,56,48,88,90,86,49,89,54,22,57,58,95,59,50,85,91,60,92,80,79,96,93,97,20,52,36,51,53,44)(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);(1,2,8,28,49,94,92,83,4,17,51,95,42,40,86,89,24,59,43,74,73,60,6,22,61,91,88,14,3,13,9,25,54,19,11,36,38,82,39,85,87,64,32,76,26,70,10,34,79,90,18,33,77,69,66,75,23,65,46,84,62,67,57,35,5,20,55,31,41,12,37,80,47,81,52,21,56,29,30,15,44,68,97,96,71,78,53,58,50,16,45,48,63,93,72,27,7)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^28,h^b=h^8,h^c=h^64,h^d=h^22,h^e=h^96140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21809);