Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C151 · Schur multiplier (H2) = C24 |
| direct product | C151×#13315 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C151 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C151 |
| prime radical | C2×#41×C151 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C151 ◁ C22×C4×C151 ◁ C2×#131×C151 ◁ G | |
516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 219 |
| transitive degree | μt(G) = 19328 |
A minimal faithful permutation representation — 3 generators on 219 points:
(1,2)(3,4)(5,6,12,18)(7,19,24,50)(8,25,29,47)(9,30,34,45)(10,17,11,16)(13,36,40,21)(14,35,43,28)(15,32,46,33)(20,51,53,68)(22,44,23,49)(26,48,55,66)(27,58,61,42)(31,38,54,39)(37,63,64,57)(41,59,65,52)(56,62,67,60)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219);(5,7,12,24)(6,13,18,40)(8,26,29,55)(9,31,34,54)(10,23,11,22)(14,41,43,65)(15,44,46,49)(16,39,17,38)(19,45,50,30)(20,35,53,28)(21,32,36,33)(25,37,47,64)(27,59,61,52)(42,66,58,48)(51,67,68,56)(57,62,63,60);(2,4)(5,8)(6,14)(7,20)(9,27)(10,35)(11,28)(12,29)(13,37)(15,42)(16,47)(17,25)(18,43)(19,48)(21,52)(22,55)(23,26)(24,53)(30,56)(31,57)(32,62)(33,60)(34,61)(36,59)(38,65)(39,41)(40,64)(44,51)(45,67)(46,58)(49,68)(50,66)(54,63)6040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 6303);