Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C107 · Schur multiplier (H2) = C2×C4 |
| direct product | C107×#8279 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C107 | C2×D4 |
| commutator | C2×C4 | C2×C42×C107 |
| Frattini, cosocle | C2×C42 | C23×C107 |
| socle, prime radical | C23×C107 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C107 ◁ C43×C107 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 27392 |
A minimal faithful permutation representation — 3 generators on 135 points:
(1,2,4,3)(5,6,8,10,9,11,7,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,24)(23,26)(27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(6,10)(7,8)(11,12)(13,15,16,21,17,22,23,27)(14,18,20,25,19,24,26,28);(1,3,4,2)(6,11)(10,12)(14,19)(18,24)(20,26)(25,28)8132 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 1267);