Finite Group DatabasebetaFinite Group Databasebeta| order | 20096 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C157 · Schur multiplier (H2) = C22 |
| direct product | #2170×C157 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C157 | D4 |
| commutator | C4 | C22×C8×C157 |
| Frattini, cosocle | C2×C8 | C23×C157 |
| socle | C2×C157 | C8×D4 |
| prime radical | #135×C157 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C157 ◁ C2×C16×C157 ◁ G | |
146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,6,11,5,12,22,30,7,13,24,15,23,10,28,29)(3,14,17,31,18,26,32,4,9,20,27,19,16,8,25,21)(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)(2,8)(4,15)(5,18)(6,17)(7,9)(10,20)(11,21)(12,14)(13,26)(16,23)(19,29)(22,32)(24,27)(25,28)(30,31);(1,4,6,20,5,19,22,8,7,21,24,14,23,31,28,26)(2,9,11,27,12,16,30,25,13,3,15,17,10,18,29,32)8792 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 902);