Finite Group DatabasebetaFinite Group Databasebeta| order | 12978 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(ΓL(1,𝔽8)⋊D103) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D309 |
| Fitting, socle, derived | C3×C7×C103 ⋊ C2×C3 |
| Hall | C3×ΓL(1,𝔽8)×C103 ⋊ C2 |
| Hall | C7×C103 ⋊ C3×D3 |
| Sylow | C103 ⋊ C3⋊AGL(1,𝔽7) |
| Hall | C3×ΓL(1,𝔽8) ⋊ D103 |
| Sylow | C7 ⋊ C3×D309 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C103 | C2×C3 |
| LCS | G ▷ C3×C7×C103 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 2163 |
A minimal faithful permutation representation — 2 generators on 113 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)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(105,108,109,106,107,110)(112,113);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(104,105,107,110,109,108,106)(111,112,113)Polycyclic presentation:
2,3^2,7,103|c^a=c^2,d^a=d^6,e^a=e^102,d^b=d^2777 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12978, 49);