Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C107 · Schur multiplier (H2) = C24 |
| direct product | C2×C107×#1331 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C107 | D8 |
| commutator | C8 | C22×C8×C107 |
| Frattini, cosocle | C4×C8 | C23×C107 |
| socle | C23×C107 | #102 |
| prime radical | C22×D8×C107 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C107 ◁ C2×C42×C107 ◁ C2×C4×C8×C107 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 3 generators on 133 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);(3,4,5,7,6,8,9,10)(11,12)(13,16)(14,23)(15,19)(17,20)(18,21)(22,25)(24,26);(12,16)(13,20)(14,21)(17,18)(19,25)(22,26)(23,24)9416 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 4577);