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 | C32 | #87604 |
| commutator, hypocenter | C97 | C128 |
| Fitting | C32×C97 | C4 |
| socle, prime radical | C2×C97 | C64 |
| cosocle | #20260454 | C2 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C32 | |
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:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95)(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);(1,2,5,10,19,32,49,70,64,45,62,80,72,47,67,85,92,83,63,42,25,12,3,7,14,23,36,53,74,60,43,28,41,58,78,68,87,94,96,90,81,59,38,21,8,16,27,40,57,77,56,39,26,15,24,37,54,75,89,95,97,93,86,79,55,34,17,29,44,61,73,52,35,22,13,6,11,20,33,50,71,88,91,84,66,82,76,51,30,46,65,69,48,31,18,9,4)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^22,h^b=h^96896 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19325);