Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×C53×#1287 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #95 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C24×C53 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 87 points:
(1,2)(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);(3,4,6,11,8,10,7,12)(5,13,14,18,15,9,16,17)(19,20)(21,27,29,33)(22,25)(23,26)(24,28,31,34)(30,32);(3,5,7,14,8,15,6,16)(4,9,12,17,10,13,11,18)(19,21)(20,24,26,31)(22,28)(23,29)(25,27,32,33)(30,34)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 951);