Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C4 |
| direct product | C53×#13113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C4×D8 |
| prime radical | C23×C53 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,19,13,18,30,20)(11,21,14,31,23,16,29,34)(12,24,27,33,25,40,28,39)(17,35,37,22,36,32,38,26)(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);(2,5)(6,8)(9,11,15,14,13,23,30,29)(10,16,19,34,18,21,20,31)(12,22,27,32,25,26,28,35)(17,33,37,40,36,39,38,24);(1,3,4,7)(2,6,5,8)(9,12,13,25)(10,17,18,36)(11,22,23,26)(14,32,29,35)(15,27,30,28)(16,33,21,39)(19,37,20,38)(24,34,40,31)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6101);