Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Sylow | #3483 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | #3483 | C3 |
| hypercenter | C4×Q8 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
292 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,4,5,3)(2,6,7,8)(9,10,13,16)(11,15,19,22,12,21,18,24,17,14,20,23);(1,3,5,4)(2,6,7,8)(9,17,13,19)(10,21,16,23)(11,20,18,12)(14,24,22,15);(1,2,5,7)(3,8,4,6)Polycyclic presentation:
2^3,3,2^4|a^2=e,b^2=e,c^2=eh,f^2=h,g^2=h,b^a=be,f^d=g,g^d=fg,g^f=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #17520⋊C3, #11710305, #60328⋊C3, … |
| maximal normal subgroup of | #17520⋊C3, #11710305, #60328⋊C3, … |
| maximal quotient of | #34129⋊C3, C4×C4⋊(#41⋊C3), C4×Q8×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18180);