Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C97 · Schur multiplier (H2) = C24 |
| direct product | C2×#479×C97 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C97 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C97 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C97 ◁ C2×C42×C97 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(3,4,6,8)(5,9,10,7)(11,12,14,16)(13,19,20,26)(15,23,22,18)(17,24,25,21);(3,5,6,10)(4,7,8,9)(12,16)(17,25)(18,23)(19,26);(11,13,14,20)(12,17,16,25)(15,24,22,21)(18,26,23,19)3686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 1807);