Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C149 · Schur multiplier (H2) = C25 |
| direct product | C2×#510×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C149 |
| prime radical | C2×#135×C149 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C22×C4×C149 ◁ C22×C8×C149 ◁ G | |
360 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 183 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 4 generators on 183 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183);(3,4)(5,13)(6,17)(7,22)(8,10)(9,14)(11,20)(12,19)(15,33)(16,30)(18,28)(21,29)(23,27)(24,32)(25,34)(26,31);(3,5,8,16)(4,9,10,24)(6,18,21,34)(7,23,19,15)(11,27,26,33)(12,28,22,25)(13,29,30,17)(14,31,32,20);(4,10)(9,24)(11,26)(12,22)(13,30)(17,29)(25,28)(27,33)4768 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#13023, C149×#13026, C149×#13028, … |
| maximal normal subgroup of | C149×#13023, C149×#13026, C149×#13028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 2145);