Finite Group DatabasebetaFinite Group Databasebeta| order | 13132 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | C7⋊Dic469 |
| unambiguous semidirect product | C67⋊(C7⋊Dic7) |
| derived, Hall | C72×C67 ⋊ C4 |
| Sylow | C67 ⋊ C7⋊Dic7 |
| Sylow | C72 ⋊ Dic67 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊D469 |
| commutator, hypocenter | C72×C67 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C72×C67 | C2 |
| LCS | G ▷ C72×C67 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13132 |
A minimal faithful permutation representation — 3 generators on 85 points:
(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69,70,71)(73,74)(75,76)(77,78)(80,81)(82,83)(84,85);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(79,80,82,84,85,83,81);(72,73,75,77,78,76,74)Polycyclic presentation:
2,7^2,67,2|a^2=e,b^a=b^6,c^a=c^6,d^a=d^663286 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13132, 15);