Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C2×C4 |
| direct product | #1521×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C149 |
| Frattini, cosocle | C22×C4 | C23×C149 |
| prime radical | #121×C149 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C2×C42×C149 ◁ G | |
218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 218 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:
(2,7,9,21)(3,12)(4,17)(8,23,24,29)(10,22,11,18)(13,28,25,20)(15,31)(16,27)(19,30)(26,32)(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);(1,2)(3,13)(4,18)(5,10)(6,11)(7,19)(8,16)(9,14)(12,24)(15,28)(17,21)(20,31)(22,30)(23,32)(25,27)(26,29);(1,3,14,27)(2,8,9,24)(4,15,30,31)(5,16,6,12)(7,20,21,28)(10,13,11,25)(17,32,19,26)(18,23,22,29)4768 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#8500, C149×#8501, C149×#8502, … |
| maximal normal subgroup of | C149×#8500, C149×#8501, C149×#8502, … |
| maximal quotient of | C149×#7580, C149×#7581, C149×#7584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 253);