Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C149 · Schur multiplier (H2) = C22 |
| direct product | #1301×C149 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C149 | C22⋊C4 |
| commutator | C2×C4 | C42×C149 |
| Frattini, cosocle | C2×C42 | C22×C149 |
| socle | C22×C149 | #95 |
| prime radical | C23×C149 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C149 ◁ C2×C42×C149 ◁ G | |
148 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 2 generators on 173 points:
(2,6)(3,4)(7,8)(9,10,12,15,13,16,20,23)(11,17,19,24,18,14,21,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(1,2,4,7,5,8,3,6)(9,11)(10,14,16,17)(12,19)(13,18)(15,22,23,24)(20,21)5662 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#1301×C149, C149×#8076, C149×#8077, … |
| maximal normal subgroup of | C2×#1301×C149, C149×#8076, C149×#8077, … |
| maximal quotient of | C149×#7027, C149×#7029, C149×#7031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 33);