Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C97 · Schur multiplier (H2) = C25 |
| direct product | C97×#1795 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C97 | #379 |
| commutator | C2×C4 | C22×C4×C97 |
| Frattini, cosocle | C22×C4 | C23×C97 |
| prime radical | C2×H(5,𝔽2)×C97 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C97 ◁ C22×C4×C97 ◁ G | |
588 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 1552 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)2813 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 527);