Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | C22×H(5,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Fitting | C22×H(5,𝔽2) | C3 |
| socle | C23 | C22×A4 |
| hypercenter | Q8 | C22⋊A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
1155 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1155 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 24 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,11,20,6,4,19,7,17,12,3,15,24)(2,14,21,5,9,22,10,16,18,8,13,23);(1,2,7,10)(3,5,6,8)(4,13,15,9)(11,14,17,16)(12,21,20,18)(19,22,24,23);(1,10,7,2)(3,5,6,8)(4,11,15,17)(9,16,13,14)(12,23,20,22)(18,24,21,19)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,d^2=h,e^2=h,b^a=bh,d^c=eh,e^c=de,f^c=g,g^c=fg,e^d=eh36 conjugacy classes — view the full character table.
| maximal subgroup of | #32884⋊C3, #32886⋊C3, #11709943, … |
| maximal normal subgroup of | #32884⋊C3, #32886⋊C3, #11709943, … |
| maximal quotient of | #11710594, C4⋊#5948, Q8×C22⋊SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20119);