Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22 |
| direct product | D53×M128 |
| Fitting | C53×M128 ⋊ C2 |
| Sylow | C53 ⋊ C2×M128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C22×D53 |
| commutator, socle | C2×C53 | C22×C32 |
| Fitting | C53×M128 | C2 |
| prime radical | C22×D53 | C32 |
| cosocle | C32×C53 | C23 |
| hypercenter | M128 | D53 |
| hypocenter | C53 | C2×M128 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C32 ◁ M128 | |
134 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(2,8)(9,23)(10,24)(11,25)(12,26)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(49,58)(50,59)(51,60)(52,61)(62,64)(65,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67);(1,2,3,9,4,10,13,27,5,11,14,28,17,30,33,49,6,12,15,29,18,31,34,50,20,32,36,51,39,52,53,62,7,8,16,23,19,24,35,43,21,25,37,44,40,46,54,58,22,26,38,45,41,47,55,59,42,48,56,60,57,61,63,64);(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)2240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90603);