Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#9000 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C109 | #120 |
| commutator | C42 | C22×C4×C109 |
| Frattini, cosocle | C2×C42 | C23×C109 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C2×C42×C109 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 27904 |
A minimal faithful permutation representation — 3 generators on 137 points:
(2,4)(6,9,10,12)(13,14,17,21,19,22,18,27)(15,23,24,28,16,25,26,20)(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);(1,2,3,4)(7,11)(9,12)(13,15,19,16)(14,20,22,28)(17,26,18,24)(21,25,27,23);(5,6)(7,12)(8,10)(9,11)(13,16,19,15)(14,20,22,28)(17,26,18,24)(21,23,27,25)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 1988);