Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C149 · Schur multiplier (H2) = C26 |
| direct product | C2×#454×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C149 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C22×C4×C149 ◁ G | |
728 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 4768 |
A minimal faithful permutation representation — 4 generators on 167 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(4,8)(6,13)(9,17)(10,14)(11,15)(16,18);(3,4)(5,9)(6,14)(7,10)(8,13)(11,16)(12,18)(15,17);(3,5,7,12)(4,9,10,18)(6,15,13,11)(8,16,14,17)4768 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#13055, C149×#13057, C149×#13357, … |
| maximal normal subgroup of | C149×#13055, C149×#13057, C149×#13357, … |
| maximal quotient of | C2×#1597×C149, C2×#1615×C149, C2×#1598×C149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 1748);