Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | D53×H(7,𝔽2) |
| Fitting | C53×H(7,𝔽2) ⋊ C2 |
| Sylow | C53 ⋊ C2×H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C26×D53 |
| commutator, socle, cosocle | C2×C53 | C27 |
| Fitting | C53×H(7,𝔽2) | C2 |
| hypercenter | H(7,𝔽2) | D53 |
| hypocenter | C53 | C2×H(7,𝔽2) |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ H(7,𝔽2) | |
60570 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.
This group has 60570 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) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 7 generators on 69 points:
(2,8)(6,9)(7,14)(13,15)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(3,11)(6,9)(10,16)(13,15);(4,12)(7,14)(10,16)(13,15);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16);(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(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)1820 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1083463);