Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C151 · Schur multiplier (H2) = C22 |
| direct product | C151×#13698 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C151 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C151 |
| prime radical | C23×C151 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C151 ◁ C22×C4×C151 ◁ C22×C8×C151 ◁ C22×C16×C151 ◁ G | |
190 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 74 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)(5,6,11,16)(7,17,19,40)(8,23,26,55)(9,27,29,57)(10,30,28,15)(12,31,32,20)(13,35,37,18)(14,38,39,24)(21,51,47,68)(22,52,48,43)(25,56,45,54)(33,63,62,67)(34,50,58,61)(36,46,53,64)(41,60,65,44)(42,59,66,49)(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);(5,7,18,44,9,21,45,61,28,48,64,63,38,59,26,31,11,19,35,60,29,47,25,50,10,22,46,67,24,49,8,20)(6,12,23,42,14,33,53,52,30,58,56,68,57,65,37,40,16,32,55,66,39,62,36,43,15,34,54,51,27,41,13,17);(2,4)(7,19)(12,32)(17,40)(20,31)(21,47)(22,48)(33,62)(34,58)(41,65)(42,66)(43,52)(44,60)(49,59)(50,61)(51,68)(63,67)6946 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 6686);