Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C149 · Schur multiplier (H2) = C23 |
| direct product | #2047×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C149 |
| prime radical | C23×C149 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C22×C4×C149 ◁ G | |
256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 256 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) = 185 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 185 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,30)(8,19,22,12)(9,23,25,13)(11,24,27,17)(16,32,20,31)(21,29,36,34)(26,33,28,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(5,7,10,18)(6,11,14,27)(8,20,22,16)(9,24,25,17)(12,28,19,26)(13,30,23,15)(21,33,36,35)(29,31,34,32);(2,4)(5,8,9,21,10,22,25,36)(6,12,13,29,14,19,23,34)(7,16,17,33,18,20,24,35)(11,26,15,31,27,28,30,32)3874 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#12167, C149×#12168, C149×#12169 |
| maximal normal subgroup of | C149×#12167, C149×#12168, C149×#12169 |
| maximal quotient of | C149×#10446, C149×#10447, C149×#10454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 779);