Finite Group DatabasebetaFinite Group Databasebeta| order | 30324 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Hall | C7×C192 ⋊ C3×C4 |
| Hall | C19×Dic133 ⋊ C3 |
| Hall | #21439062 ⋊ C4 |
| Sylow | C192 ⋊ ΓL(1,𝔽8)⋊C4 |
| Sylow | C7 ⋊ #16864560 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #29503108 |
| commutator, hypocenter | C7×C192 | C3×C4 |
| Fitting, socle, cosocle | C2×C7×C192 | C2×C3 |
| prime radical | C2×#21439062 | C2 |
| LCS | G ▷ C7×C192 | |
| UCS | C1 ◁ C2 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 532 |
A minimal faithful permutation representation — 2 generators on 49 points:
(1,2,3,4)(6,9,10,7,8,11)(13,17,18,14,15,19)(16,25,22,20,21,27)(23,28,30,29,26,24)(32,33,35)(34,37,38)(36,40,39)(41,45,43)(42,46,44)(47,49,48);(5,6,8,11,10,9,7)(12,13,16,22,28,25,29,15,19,24,30,18,17,23,21,26,27,20,14)(31,32,34,38,43,48,45,33,36,41,46,35,39,44,37,42,47,49,40)Polycyclic presentation:
2,3,7,19^2,2|a^2=f,c^a=c^6,d^a=d^18,c^b=c^2,d^b=d^7,e^b=e^7872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30324, 83);