Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13655 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | D16 |
| commutator | C16 | C22×C4×C53 |
| Frattini, cosocle | C2×C16 | C23×C53 |
| socle | C2×C53 | C4×D16 |
| prime radical | C53×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C2×C8×C53 ◁ C4×C8×C53 ◁ C4×C16×C53 ◁ G | |
186 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 3 generators on 117 points:
(1,2,6,12,8,15,30,14)(3,16,19,49,22,50,21,56)(4,10,7,31,24,13,33,40)(5,25,27,57,29,39,61,58)(9,34,36,54,38,53,47,18)(11,41,43,62,45,23,44,28)(17,48,32,42,20,55,52,64)(26,46,59,63,51,35,60,37)(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)(2,9)(4,17)(5,26)(6,19)(7,32)(8,22)(10,35)(11,42)(12,36)(13,46)(14,47)(15,38)(16,25)(18,28)(20,24)(21,30)(23,53)(27,59)(29,51)(31,37)(33,52)(34,41)(39,50)(40,63)(43,55)(44,48)(45,64)(49,57)(54,62)(56,58)(60,61);(1,4,6,7,8,24,30,33)(2,10,12,31,15,13,14,40)(3,17,19,32,22,20,21,52)(5,23,27,28,29,41,61,62)(9,35,36,37,38,46,47,63)(11,39,43,58,45,25,44,57)(16,48,49,42,50,55,56,64)(18,51,34,60,54,26,53,59)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6643);