Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C26 |
| direct product | C107×#33546 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C107 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C107 |
| socle | C2×C107 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C107 ◁ C2×C4×C107 ◁ C2×#41×C107 ◁ G | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 1712 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(2,6)(7,12)(8,13)(14,16);(4,11)(9,15);(1,3,5,10)(2,7,8,14)(4,9,11,15)(6,12,13,16)4280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26534);