Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#7439 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #353 |
| commutator | C2×M16 | C2×C4×C53 |
| Frattini, cosocle | #408 | C22×C53 |
| socle | C2×C53 | #1352 |
| prime radical | C2×#41×C53 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C53×#408 ◁ G | |
170 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 2 generators on 117 points:
(1,2,5,11,8,14,7,13)(3,15,17,48,20,50,18,49)(4,21,23,58,26,45,25,47)(6,29,31,43,32,12,28,46)(9,33,34,60,36,22,44,59)(10,37,39,24,40,61,42,30)(16,51,52,41,54,64,53,35)(19,38,55,57,27,62,56,63)(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);(1,3,6,17)(2,9,29,34)(4,22,24,59)(5,27,28,56)(7,19,31,55)(8,20,32,18)(10,38,45,63)(11,41,46,64)(12,44,14,36)(13,35,43,51)(15,47,49,42)(16,37,52,23)(21,57,40,62)(25,54,61,53)(26,33,30,60)(39,50,58,48)1643 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 427);