Finite Group DatabasebetaFinite Group Databasebeta| order | 27008 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C211 · Schur multiplier (H2) = C24 |
| direct product | #3119×C211 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C211 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C211 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ #117×C211 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 235 |
| transitive degree | μt(G) = 13504 |
A minimal faithful permutation representation — 4 generators on 235 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,22)(19,21)(23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235);(2,5)(6,8)(9,11,13,18)(10,14,17,22)(12,19,20,24)(15,21,16,23);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(1,3,4,7)(2,6,5,8)(11,18)(14,22)(19,24)(21,23)6752 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27008, 1851);