Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C109 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C109×#2234 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C109 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C109 |
| prime radical | C2×#466×C109 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C23×C4×C109 ◁ C23×C8×C109 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(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);(3,4)(5,8)(6,7)(9,10)(12,16)(13,19)(15,22)(17,25)(18,21)(20,26)(23,24);(4,7)(8,10)(11,12,14,16)(13,17,20,24)(15,18,21,22)(19,23,26,25);(3,5,6,9)(4,8,7,10)(11,13,14,20)(12,17,16,24)(15,23,21,25)(18,26,22,19)6322 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 26629);