Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C149 · Schur multiplier (H2) = C26 |
| direct product | C2×#458×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C149 |
| prime radical | C2×H(5,𝔽2)×C149 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C24×C149 ◁ G | |
794 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 794 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);(3,4,7,10)(5,8,12,15)(6,9,13,16)(11,17,18,14);(3,5)(4,8)(6,11)(7,12)(9,14)(10,15)(13,18)(16,17);(3,6)(4,9)(5,11)(7,13)(8,14)(10,16)(12,18)(15,17)4768 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#13042, C149×#13044, C149×#13344, … |
| maximal normal subgroup of | C149×#13042, C149×#13044, C149×#13344, … |
| maximal quotient of | C2×#1607×C149, C2×#1613×C149, C2×#1608×C149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 1756);