Finite Group DatabasebetaFinite Group Databasebeta| order | 12608 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C197 · Schur multiplier (H2) = C23 |
| direct product | #421×C197 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C197 | C2×D4 |
| commutator | C4 | C22×C4×C197 |
| Frattini, cosocle | C2×C4 | C23×C197 |
| socle | C2×C197 | C2×C22⋊C4 |
| prime radical | C2×#41×C197 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C197 ◁ C22×C4×C197 ◁ G | |
154 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 3152 |
A minimal faithful permutation representation — 3 generators on 213 points:
(1,2,5,8,6,9,4,13)(3,10,12,7,11,16,14,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13);(3,11)(7,15)(10,16)(12,14)4334 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12608, 102);