Finite Group DatabasebetaFinite Group Databasebeta| order | 20096 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C157 · Schur multiplier (H2) = C2×C4 |
| direct product | #1515×C157 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C157 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C157 |
| Frattini, cosocle | C22×C4 | C23×C157 |
| prime radical | C23×C157 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C157 ◁ C2×C42×C157 ◁ G | |
208 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 189 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31)(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);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,23)(22,31)(24,28)(25,32)4082 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 247);