Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C109 · Schur multiplier (H2) = C22 |
| direct product | C109×#7484 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #1390 |
| commutator | C2×Q8 | C42×C109 |
| Frattini, cosocle | #532 | C22×C109 |
| prime radical | #532×C109 | C22 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C22×C4×C109 ◁ #532×C109 ◁ G | |
266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 2 generators on 141 points:
(1,2,4,9,6,12,21,28)(3,13,15,7,18,23,31,25)(5,11,19,8,22,10,20,26)(14,29,17,27,32,24,16,30)(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);(1,3,5,16,6,18,22,17)(2,7,10,24,12,25,11,29)(4,14,19,31,21,32,20,15)(8,13,9,27,26,23,28,30)3052 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 472);