Finite Group DatabasebetaFinite Group Databasebeta| order | 30492 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2×C11 |
| unambiguous semidirect product | D7⋊((C11⋊D11)⋊C9) |
| unambiguous semidirect product | #29503838⋊C2 |
| unambiguous semidirect product | (D7⋊(C112⋊C9))⋊C2 |
| unambiguous semidirect product | ((C11⋊D11)⋊(C7⋊C9))⋊C2 |
| derived, Hall | C7×C112 ⋊ C22×C9 |
| Hall | C7⋊(C112⋊C9) ⋊ C22 |
| Hall | D7×C11⋊D11 ⋊ C9 |
| Sylow | C112 ⋊ C2×D7⋊C9 |
| Sylow | C7 ⋊ C2×(C11⋊D11)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | D7⋊((C11⋊D11)⋊C3) |
| commutator, hypocenter | C7×C112 | C22×C9 |
| Fitting, socle, cosocle | C3×C7×C112 | C22×C3 |
| prime radical | C3×D7×C11⋊D11 | C3 |
| LCS | G ▷ C7×C112 | |
| UCS | C1 ◁ C3 | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 693 |
A minimal faithful permutation representation — 2 generators on 49 points:
(2,5,6,3,4,7)(8,12,16,10,15,11,13,9,14)(17,21,32)(18,26,46)(19,34,20)(22,41,37)(23,31,44)(24,40,49)(25,33,27)(28,29,45)(30,39,42)(35,43,38)(36,47,48);(1,2,4,7,6,5,3)(19,23)(21,28)(22,24)(25,35)(26,38)(27,32)(29,40)(30,34)(31,43)(33,36)(37,45)(39,47)(41,44)(42,46)(48,49)Polycyclic presentation:
2^2,3,7,11^2,3|c^3=g,e^a=e^10,f^a=f^10,d^b=d^6,d^c=d^2,e^c=e^3f^9,f^c=ef^7342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30492, 149);