Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C22 |
| direct product | C149×#13657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C149 | #2177 |
| commutator | C16 | C22×C4×C149 |
| Frattini, cosocle | C2×C16 | C23×C149 |
| prime radical | #508×C149 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C149 ◁ C2×C4×C149 ◁ C4×C8×C149 ◁ #346×C149 ◁ G | |
182 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,15,18,49)(4,12)(5,23)(6,29)(7,32)(8,14)(9,17,35,56)(10,21)(11,26)(13,31)(16,51,53,64)(19,58,33,50)(20,59)(22,44)(24,43,52,48)(25,41)(27,38)(28,46)(30,47,55,39)(34,62,36,63)(37,61,45,57)(40,60)(42,54)(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)(2,9)(4,16)(5,24)(6,30)(7,33)(8,18)(10,34)(11,39)(12,43)(13,45)(14,35)(15,46)(17,54)(19,31)(20,57)(21,55)(22,53)(23,49)(25,61)(26,56)(27,63)(28,52)(29,36)(32,37)(38,58)(40,50)(41,51)(42,47)(44,48)(59,64)(60,62);(1,4,7,21,8,22,31,6)(2,10,13,12,14,29,32,44)(3,16,33,55,18,53,19,30)(5,25,27,26,28,20,60,54)(9,34,45,43,35,36,37,48)(11,40,41,46,42,38,59,23)(15,47,58,64,49,39,50,51)(17,24,61,63,56,52,57,62)5066 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6645);