Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C23 |
| direct product | #2135×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C149 | C2×#99 |
| commutator | C2×C4 | C22×C4×C149 |
| Frattini, cosocle | C2×Q8 | C23×C149 |
| prime radical | C2×#41×C149 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C149 ◁ C2×C4×C149 ◁ C2×#41×C149 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 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) = 181 |
| transitive degree | μt(G) = 4768 |
A minimal faithful permutation representation — 3 generators on 181 points:
(1,2,6,11,7,12,4,9)(3,13,16,31,17,28,14,30)(5,19,18,29,21,10,22,27)(8,23,24,20,25,32,26,15)(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);(1,3,7,17)(2,8,12,25)(4,16,6,14)(5,20,21,15)(9,26,11,24)(10,28,19,13)(18,23,22,32)(27,30,29,31);(1,4,7,6)(2,9,12,11)(3,14,17,16)(5,18,21,22)(8,24,25,26)(10,27,19,29)(13,30,28,31)(15,23,20,32)3427 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C149×#12712, C149×#12713, C149×#12721, … |
| maximal normal subgroup of | C149×#12712, C149×#12713, C149×#12721, … |
| maximal quotient of | C149×#11434, C149×#11450, C149×#11481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 867);