Finite Group DatabasebetaFinite Group Databasebeta| order | 20096 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C157 · Schur multiplier (H2) = C22 |
| direct product | #1395×C157 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C157 | #99 |
| commutator | C2×C4 | C42×C157 |
| Frattini, cosocle | C2×#41 | C22×C157 |
| socle | C2×C157 | #342 |
| prime radical | C2×#41×C157 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C157 ◁ C2×C4×C157 ◁ C2×#41×C157 ◁ G | |
162 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 189 |
| transitive degree | μt(G) = 5024 |
A minimal faithful permutation representation — 2 generators on 189 points:
(1,2,5,10)(3,13,15,29)(4,11,18,27)(6,9,20,26)(7,12,21,28)(8,22,23,14)(16,25,30,32)(17,24,19,31)(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);(1,3,6,16,7,17,4,14)(2,8,11,24,12,25,9,13)(5,19,20,22,21,15,18,30)(10,23,27,31,28,32,26,29)4082 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 127);