Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×SD128 ⋊ C2 |
| Sylow | C53 ⋊ #13742 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×D53 |
| commutator, cosocle | C32×C53 | C23 |
| Frattini | C32 | C22×D53 |
| Fitting | C53×SD128 | C2 |
| socle | C2×C53 | C2×D32 |
| hypercenter | SD128 | D53 |
| hypocenter | C53 | #13742 |
| LCS | G ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ SD128 | |
134 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 39 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)(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)(55,62)(61,80)(63,84)(64,85)(65,86)(66,79)(72,103)(76,109)(81,111)(82,112)(83,91)(87,113)(88,110)(89,104)(90,114)(105,107);(55,61)(56,67)(57,68)(58,73)(59,77)(62,80)(63,64)(65,90)(66,91)(69,99)(70,102)(71,95)(72,81)(74,108)(75,98)(76,82)(78,106)(79,83)(84,85)(86,114)(87,105)(88,104)(89,110)(92,93)(94,100)(96,97)(101,117)(103,111)(107,113)(109,112)(115,116);(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)(54,55)(56,61)(57,72)(58,76)(59,79)(60,62)(63,67)(64,68)(65,73)(66,77)(69,90)(70,91)(71,80)(74,107)(75,103)(78,109)(81,93)(82,92)(83,100)(84,95)(85,98)(86,106)(87,99)(88,102)(89,108)(94,110)(96,105)(97,104)(101,114)(111,116)(112,115)(113,117)Polycyclic presentation:
2^8,53|d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=de,e^a=ef,f^a=fg,g^a=gh,i^a=i^52,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^c=i^52956 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90627);