Finite Group DatabasebetaFinite Group Databasebeta| order | 36693 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C151 · Schur multiplier (H2) = C1 |
| direct product | C151×C81⋊C3 |
| unambiguous semidirect product | C81⋊(C3×C151) |
| subgroup | G/H | |
|---|---|---|
| center | C27×C151 | C32 |
| commutator | C3 | C3×C27×C151 |
| Frattini, cosocle | C27 | C32×C151 |
| socle | C3×C151 | C3×C27 |
| prime radical | C32×C151 | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C27×C151 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 232 |
| transitive degree | μt(G) = 12231 |
A minimal faithful permutation representation — 2 generators on 232 points:
(1,2,6,3,8,18,11,23,37,4,9,19,12,24,38,28,45,59,14,26,39,30,47,60,50,66,74,5,10,17,13,25,35,29,46,56,15,27,36,31,48,57,51,67,72,33,49,58,53,68,73,69,78,80,16,7,20,32,21,40,52,42,61,34,22,41,54,43,62,70,64,75,55,44,63,71,65,76,79,77,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232);(2,7,10)(6,17,20)(8,21,25)(9,22,27)(18,35,40)(19,36,41)(23,42,46)(24,43,48)(26,44,49)(37,56,61)(38,57,62)(39,58,63)(45,64,67)(47,65,68)(59,72,75)(60,73,76)(66,77,78)(74,80,81)14949 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36693, 24);