Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
No aliases for this group.
| Fitting | C53×2−1+6 ⋊ C2 |
| Sylow | C53 ⋊ #63103 |
| 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 | #63103 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ 2−1+6 | |
59452 conjugacy classes of subgroups; only the 32039 normal subgroups are shown.
This group has 59452 conjugacy classes of subgroups — too many to draw. Only its 32039 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:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(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)(54,58,59,70)(55,63,62,75)(56,67,66,80)(57,69,68,81)(60,72,64,77)(61,74,73,83)(65,79,78,85)(71,82,76,84);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(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)(55,62)(60,64)(61,73)(63,75)(71,76)(72,77)(74,83)(82,84);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(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)(56,66)(60,64)(65,78)(67,80)(71,76)(72,77)(79,85)(82,84);(57,68)(61,73)(65,78)(69,81)(71,76)(74,83)(79,85)(82,84);(54,57)(55,61)(56,65)(58,69)(59,68)(60,71)(62,73)(63,74)(64,76)(66,78)(67,79)(70,81)(72,82)(75,83)(77,84)(80,85);(54,56)(55,60)(57,65)(58,67)(59,66)(61,71)(62,64)(63,72)(68,78)(69,79)(70,80)(73,76)(74,82)(75,77)(81,85)(83,84);(54,55)(56,64)(57,61)(58,63)(59,62)(60,66)(65,76)(67,77)(68,73)(69,74)(70,75)(71,78)(72,80)(79,84)(81,83)(82,85)Polycyclic presentation:
2^8,53|g^2=h,b^a=bh,f^a=fh,e^b=eh,d^c=dh,i^e=i^52,i^f=i^52,i^g=i^521820 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1083471);