Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | H(7,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Fitting | H(7,𝔽2) | C3 |
| hypercenter | D4 | C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ D4 | |
1077 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1077 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6)(3,9,12)(4,11,10)(7,16,14,13,15,8);(1,2,5,6)(3,13,11,7)(4,14,12,8)(9,15,10,16);(1,9)(2,15)(3,12)(4,11)(5,10)(6,16)(7,14)(8,13)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^a=bh,d^c=deh,e^c=dh,f^c=fgh,g^c=fh,g^d=gh,f^e=fh35 conjugacy classes — view the full character table.
| maximal subgroup of | #32894⋊C3, #32896⋊C3, #60399⋊C3, … |
| maximal normal subgroup of | #32894⋊C3, #32896⋊C3, #60399⋊C3, … |
| maximal quotient of | #11710602, #11710604, #62843⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20123);