Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C149 · Schur multiplier (H2) = C22 |
| direct product | C16×Q16×C149 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C149 | D4 |
| commutator | C4 | C22×C16×C149 |
| Frattini, cosocle | C4×C8 | C23×C149 |
| socle, prime radical | C22×C149 | C8×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C149 ◁ C4×C16×C149 ◁ G | |
218 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 38144 |
A minimal faithful permutation representation — 3 generators on 181 points:
(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)16688 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 5040);