Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C26 |
| direct product | C2×C107×#2197 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C107 |
| prime radical | C2×B(4,𝔽2)×C107 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C23×C107 ◁ C22×D4×C107 ◁ G | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3424 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(3,4)(5,11)(6,9)(7,10)(8,12)(13,17)(14,16)(15,18);(3,5,7,13)(4,8,10,15)(6,12,14,18)(9,11,16,17);(6,14)(12,18)4280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26532);