Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C23 |
| direct product | C149×#13608 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C149 | #2127 |
| commutator | C42 | C22×C4×C149 |
| Frattini, cosocle | #124 | C23×C149 |
| prime radical | C2×#41×C149 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C149 ◁ C22×C149 ◁ C22×C4×C149 ◁ #536×C149 ◁ G | |
308 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,13,16,40)(4,9)(5,21,23,55)(6,25,27,11)(7,12)(8,15,29,46)(10,24)(14,39,43,61)(17,36,26,41)(18,49,22,54)(19,51,53,34)(20,35)(28,45,59,64)(30,47,31,48)(32,42,60,63)(33,50)(37,56)(38,57)(44,58,52,62)(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);(1,3,7,17)(2,8,12,31)(4,14,20,44)(5,15,23,48)(6,26,27,16)(9,32,35,28)(10,36,38,13)(11,29,25,30)(18,42,22,45)(19,52,53,43)(21,41,55,40)(24,46,57,47)(33,58,37,39)(34,60,51,59)(49,62,54,61)(50,63,56,64);(1,4,7,20)(2,9,12,35)(3,14,17,44)(5,22,23,18)(6,19,27,53)(8,28,31,32)(10,37,38,33)(11,34,25,51)(13,39,36,58)(15,45,48,42)(16,43,26,52)(21,54,55,49)(24,56,57,50)(29,59,30,60)(40,61,41,62)(46,64,47,63)3725 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6596);