Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | D53×2−1+6 |
| Fitting | C53×2−1+6 ⋊ C2 |
| Sylow | C53 ⋊ C2×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C26×D53 |
| commutator, socle, cosocle | C2×C53 | C27 |
| Fitting | C53×2−1+6 | C2 |
| hypercenter | 2−1+6 | D53 |
| hypocenter | C53 | C2×2−1+6 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ 2−1+6 | |
59186 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.
This group has 59186 conjugacy classes of subgroups — too many to draw. Only its 32041 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 7 generators on 85 points:
(2,10)(7,11)(8,22)(9,24)(19,25)(20,26)(21,23)(31,32)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(3,14)(7,11)(12,28)(13,30)(19,25)(20,26)(27,29)(31,32);(1,5,6,18)(2,9,10,24)(3,13,14,30)(4,15,16,17)(7,20,11,26)(8,21,22,23)(12,27,28,29)(19,31,25,32);(1,4,6,16)(2,8,10,22)(3,12,14,28)(5,17,18,15)(7,19,11,25)(9,23,24,21)(13,29,30,27)(20,32,26,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,27)(16,28)(17,29)(18,30)(21,31)(22,25)(23,32)(24,26);(1,2)(3,11)(4,8)(5,9)(6,10)(7,14)(12,25)(13,26)(15,21)(16,22)(17,23)(18,24)(19,28)(20,30)(27,32)(29,31);(34,35)(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)1820 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1083466);