Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C2×C4 |
| direct product | C149×#13633 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C149 | D16 |
| commutator | C16 | C22×C4×C149 |
| Frattini, cosocle | C2×C16 | C23×C149 |
| socle | C22×C149 | C2×D16 |
| prime radical | C23×C149 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C149 ◁ C22×C4×C149 ◁ C22×C8×C149 ◁ C22×C16×C149 ◁ G | |
190 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 217 |
| transitive degree | μt(G) = 19072 |
A minimal faithful permutation representation — 3 generators on 217 points:
(1,2,3,4)(5,6)(7,18)(8,14)(9,27)(10,31)(11,33)(12,17)(13,21)(15,19)(16,28)(20,40)(22,53)(23,55)(24,43)(25,56)(26,39)(29,59)(30,58)(32,60)(34,48)(35,62)(36,52)(37,45)(38,47)(41,44)(42,50)(46,57)(49,65)(51,61)(54,68)(63,66)(64,67)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217);(5,7,19,44,10,22,25,48,8,20,45,66,26,49,30,52,12,24,47,67,32,54,29,51,11,23,46,62,28,50,9,21)(6,13,27,42,16,35,57,55,33,61,59,68,60,64,38,43,17,36,58,65,39,63,37,40,14,34,56,53,31,41,15,18);(5,8,12,11)(6,14,17,33)(7,20,24,23)(9,25,30,29)(10,26,32,28)(13,34,36,61)(15,37,38,57)(16,31,39,60)(18,40,43,55)(19,45,47,46)(21,48,52,51)(22,49,54,50)(27,56,58,59)(35,41,63,64)(42,53,65,68)(44,66,67,62)11324 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6621);