Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting, Sylow | 2−1+6 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C24×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C3 |
| Fitting | 2−1+6 | C3 |
| hypercenter | H(5,𝔽2) | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
1216 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(4,14,5)(7,18)(9,24,10,21,23,22)(12,27,13)(15,16,17)(19,32,20,29,31,30)(25,28,26);(1,16,6,14)(2,24,8,23)(3,27,11,28)(4,5,15,17)(7,31,18,32)(9,10,21,22)(12,26,25,13)(19,30,29,20);(1,3,6,11)(2,7,8,18)(4,12,15,25)(5,13,17,26)(9,19,21,29)(10,20,22,30)(14,27,16,28)(23,31,24,32);(1,2,6,8)(3,7,11,18)(4,9,15,21)(5,10,17,22)(12,19,25,29)(13,20,26,30)(14,23,16,24)(27,31,28,32)Polycyclic presentation:
2^4,3,2^3|a^2=h,b^2=h,f^2=h,g^2=h,d^a=dh,c^b=ch,f^e=fg,g^e=f,g^f=gh67 conjugacy classes — view the full character table.
| derived subgroup of | #72242963 |
| maximal subgroup of | #60390⋊C3, #60391⋊C3, #11710349, … |
| maximal normal subgroup of | #60390⋊C3, #60391⋊C3, #11710349, … |
| maximal quotient of | #62653⋊C3, #11710605, #62848⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20113);