Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C149 · Schur multiplier (H2) = C2 |
| direct product | #1403×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C149 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C4×C149 |
| Frattini, cosocle | #124 | C22×C149 |
| prime radical | H(5,𝔽2)×C149 | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C149 ◁ C22×C149 ◁ C2×C4×C149 ◁ #124×C149 ◁ G | |
112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 165 |
| transitive degree | μt(G) = 2384 |
A minimal faithful permutation representation — 2 generators on 165 points:
(1,2,4,8,6,3,10,12)(5,13,15,11,16,9,14,7)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165);(1,3)(2,6)(4,11)(5,9)(7,10)(8,14)(12,15)(13,16)2533 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#1403×C149, C149×#13564, C149×#13565, … |
| maximal normal subgroup of | C2×#1403×C149, C149×#13564, C149×#13565, … |
| maximal quotient of | C149×#7344, C149×#7375, C149×#7409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 135);