Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C23 |
| direct product | C149×#13199 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C149 | #124 |
| commutator | C22×C4 | C22×C4×C149 |
| Frattini, cosocle | C22×C8 | C23×C149 |
| socle | C2×C149 | #1955 |
| prime radical | #575×C149 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C149 ◁ C22×C8×C149 ◁ G | |
406 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 213 points:
(1,2,6,12,7,13,28,42,8,14,29,43,5,11,16,31)(3,15,20,44,18,33,45,41,19,39,27,55,25,9,30,34)(4,10,23,37,22,36,51,60,24,38,54,63,26,40,46,57)(17,47,50,62,21,32,52,59,49,35,56,61,48,58,53,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,3,16,45,5,25,29,20,8,19,28,30,7,18,6,27)(2,9,31,44,11,39,43,34,14,33,42,55,13,15,12,41)(4,21,51,50,26,17,23,53,24,48,46,56,22,49,54,52)(10,35,60,59,40,32,37,62,38,47,57,64,36,58,63,61);(1,4)(2,10)(3,17,19,49)(5,26)(6,23)(7,22)(8,24)(9,32,33,58)(11,40)(12,37)(13,36)(14,38)(15,35,39,47)(16,46)(18,21,25,48)(20,50,27,56)(28,51)(29,54)(30,53,45,52)(31,57)(34,59,41,64)(42,60)(43,63)(44,61,55,62)7748 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6187);