Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C109 · Schur multiplier (H2) = C2 |
| direct product | C109×#7443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C4×C109 |
| Frattini, cosocle | #384 | C22×C109 |
| prime radical | C23×C109 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C23×C109 ◁ C22×C4×C109 ◁ #384×C109 ◁ G | |
198 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 2 generators on 133 points:
(2,5,6,7)(9,10)(11,18,20,23,21,24,22,14)(12,15)(13,17)(16,19)(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);(1,2)(3,7)(4,6)(5,8)(9,11,19,20,13,21,12,22)(10,14,15,18,17,23,16,24)3379 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 431);