Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C149 · Schur multiplier (H2) = C26 |
| direct product | #3022×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C149 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C149 |
| prime radical | #585×C149 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C149 ◁ C22×C4×C149 ◁ G | |
704 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 212 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 — 4 generators on 181 points:
(1,2,7,12)(3,13,16,25)(4,9,18,24)(5,11,6,10)(8,19,22,15)(14,28,31,27)(17,26,20,23)(21,29,32,30)(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)(5,15)(6,19)(7,16)(9,21)(10,13)(11,25)(12,22)(17,29)(18,31)(20,30)(23,27)(24,32)(26,28);(1,4)(2,9)(3,14)(5,17)(6,20)(7,18)(8,21)(10,23)(11,26)(12,24)(13,27)(15,29)(16,31)(19,30)(22,32)(25,28);(1,5,7,6)(2,10,12,11)(3,15,16,19)(4,17,18,20)(8,13,22,25)(9,23,24,26)(14,29,31,30)(21,27,32,28)4321 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#13355, C149×#13356, C149×#13361, … |
| maximal normal subgroup of | C149×#13355, C149×#13356, C149×#13361, … |
| maximal quotient of | C149×#18136, C149×#18140, C149×#18146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 1754);