Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C149 · Schur multiplier (H2) = C24 |
| direct product | #2009×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C149 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23×C149 |
| prime radical | #575×C149 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C149 ◁ C22×C4×C149 ◁ G | |
348 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 4768 |
A minimal faithful permutation representation — 3 generators on 181 points:
(1,2)(3,13)(4,17)(5,11)(6,10)(7,12)(8,15)(9,19)(14,22)(16,27)(18,26)(20,25)(21,23)(24,30)(28,29)(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);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,24)(10,27)(11,13)(12,23)(17,28)(18,31)(19,29)(20,30)(22,26)(25,32);(1,4,7,20)(2,9,12,26)(3,14,16,30)(5,19,6,18)(8,22,23,24)(10,17,11,25)(13,28,27,32)(15,31,21,29)3427 conjugacy classes. The character table is too large to display.
| maximal quotient of | C149×#9931, C149×#9933, C149×#9935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 741);