Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C149 · Schur multiplier (H2) = C25 |
| direct product | #3414×C149 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C149 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C149 |
| socle | C2×C149 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C149 ◁ C22×C4×C149 ◁ C22×C8×C149 ◁ G | |
364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 181 points:
(1,2)(3,12)(4,10)(5,17)(6,21)(7,9)(8,14)(11,19)(13,29)(15,28)(16,26)(18,24)(20,31)(22,27)(23,30)(25,32)(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);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,23)(10,24)(11,27)(12,17)(14,19)(16,30)(20,32)(21,25)(26,29)(28,31);(2,9)(8,23)(10,21)(11,26)(12,28)(17,31)(24,25)(27,29);(1,4,7,6)(2,10,9,21)(3,13,15,22)(5,16,20,19)(8,24,23,25)(11,17,26,31)(12,29,28,27)(14,18,30,32)4768 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#13021, C149×#13024, C149×#13027, … |
| maximal normal subgroup of | C149×#13021, C149×#13024, C149×#13027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 2146);