Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C151 · Schur multiplier (H2) = C23 |
| direct product | C151×#13564 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C151 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4×C151 |
| Frattini, cosocle | #124 | C23×C151 |
| socle | C2×C151 | C2×AΣL(1,𝔽16) |
| prime radical | #3018×C151 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C151 ◁ C2×C4×C151 ◁ C22×C4×C151 ◁ #532×C151 ◁ G | |
494 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 2416 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2,5,10,7,3,11,14)(4,8,15,9,6,12,16,13)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(1,3)(2,7)(4,12)(5,13)(6,8)(9,11)(10,15)(14,16);(1,4,7,6)(2,8,3,12)(5,15,11,16)(9,14,13,10)5134 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 6552);