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 | C7⋊(C19⋊Dic19) ⋊ C3 |
| Hall | #21439061 ⋊ C4 |
| Sylow | C192 ⋊ ΓL(1,𝔽8)⋊C4 |
| Sylow | C7 ⋊ #16864558 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #29503106 |
| commutator, hypocenter | C7×C192 | C3×C4 |
| Fitting, socle, cosocle | C2×C7×C192 | C2×C3 |
| prime radical | C2×#21439061 | C2 |
| LCS | G ▷ C7×C192 | |
| UCS | C1 ◁ C2 | |
264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 10108 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4)(6,9,10,7,8,11)(13,17,18,14,15,19)(16,24,25,20,21,29)(22,30,28,27,26,23)(32,36,37,33,34,38)(35,43,44,39,40,48)(41,49,47,46,45,42);(5,6,8,11,10,9,7)(31,32,35,40,45,48,47,37,36,41,46,34,38,42,44,49,43,39,33);(12,13,16,21,26,29,28,18,17,22,27,15,19,23,25,30,24,20,14)Polycyclic presentation:
2,3,7,19^2,2|a^2=f,c^a=c^6,d^a=d^18,e^a=e^18,c^b=c^2,d^b=d^11,e^b=e^11854 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30324, 81);