Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C149 · Schur multiplier (H2) = C24 |
| direct product | #2987×C149 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C149 | C24 |
| commutator | C22 | C23×C4×C149 |
| Frattini, cosocle | C2×C4 | C24×C149 |
| socle | C22×C149 | C23×C4 |
| prime radical | C23×C149 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C149 ◁ G | |
314 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 9536 |
A minimal faithful permutation representation — 4 generators on 181 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(19,25)(24,30)(26,31);(2,8)(6,9)(7,16)(14,15)(19,25)(22,28)(26,31)(29,32)5662 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#8919, C149×#9006, C149×#9008, … |
| maximal normal subgroup of | C149×#8919, C149×#9006, C149×#9008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 1719);