Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C4 |
| direct product | C103×#7693 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C103 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ C43×C103 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 26368 |
A minimal faithful permutation representation — 3 generators on 135 points:
(2,6,7,8)(9,10,13,15,14,11,12,16)(17,18)(19,25,27,22)(20,28)(21,24)(23,29)(26,30,32,31)(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);(1,2)(3,6)(4,8)(5,7)(9,11,14,10)(12,15,13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,15,11,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)5974 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 681);