Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12715 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #1781 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C2×#41 | C23×C109 |
| prime radical | C109×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×2−1+4×C109 ◁ G | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2)(3,12,15,25)(4,9,16,7)(5,10)(6,11)(8,14,23,18)(13,28)(17,24,31,21)(19,26)(20,30)(22,32)(27,29)(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);(2,7)(4,16)(8,20)(9,11)(10,21)(12,27)(13,29)(17,31)(22,23)(24,26)(25,28)(30,32);(1,3)(2,8)(4,13)(5,18)(6,15)(7,20)(9,22)(10,25)(11,23)(12,26)(14,19)(16,29)(17,32)(21,28)(24,27)(30,31)3706 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5703);