Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #1790⋊C3 |
| Fitting, Sylow | #1790 ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #120⋊C3 |
| commutator, hypocenter | H(5,𝔽2) | C3×C4 |
| Frattini | C24 | C2×A4 |
| Fitting | #1790 | C3 |
| prime radical | C2×#997 | C2 |
| cosocle | C2×H(5,𝔽2) | C2×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
155 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 14 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4)(6,11,12,9,10,7);(5,10)(6,12)(7,9)(8,11)Polycyclic presentation:
2,3,2^6|a^2=c,d^a=dfh,e^a=egh,d^b=deg,e^b=dfh,f^b=fg,g^b=f,e^d=eh,g^d=gh,f^e=fh32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1790⋊C3, #20608⋊C3, #20609⋊C3, … |
| maximal normal subgroup of | C2×#1790⋊C3, #20608⋊C3, #20609⋊C3, … |
| maximal quotient of | #8316⋊C3, #8317⋊C3, #11390⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 602);