Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #857⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| derived | C42⋊C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #857⋊C2 |
| commutator, hypocenter | C42⋊C3 | C2×C4 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C2×#127 | D3 |
| prime radical | C2×#127⋊C3 | C2 |
| cosocle | C2×C42⋊C3 | C22 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 | |
168 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 16 points:
(5,6,13,8,11,15)(7,16,9)(10,14,12);(1,2,3,4)(6,9)(10,15)(11,14)(13,16)Polycyclic presentation:
2^3,3,2^4|a^2=c,e^2=g,f^2=gh,d^a=d^2,e^a=fh,f^a=eh,g^a=gh,e^b=eg,f^b=fgh,e^d=fg,f^d=ef,g^d=gh,h^d=g28 conjugacy classes — view the full character table.
| maximal subgroup of | (#560⋊C3)⋊C4, #4626⋊C4, C2×#73083, … |
| maximal normal subgroup of | (#560⋊C3)⋊C4, #4626⋊C4, C2×#73083, … |
| maximal quotient of | #11711110, #11711111, #11711132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5659);