Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | 2−1+6 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Fitting | 2−1+6 | C3 |
| hypercenter | Q8 | C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
1031 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1031 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,3,4,12)(2,7,8,11)(5,14,18,27,6,29,15,28,17,13,16,30)(9,20,24,25,10,31,21,26,23,19,22,32);(1,23)(2,18)(3,32)(4,24)(5,22)(6,9)(7,29)(8,17)(10,15)(11,30)(12,31)(13,20)(14,25)(16,21)(19,28)(26,27);(5,15)(6,16)(9,21)(10,22)(13,27)(14,28)(19,25)(20,26)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,f^2=h,g^2=h,b^a=bh,d^c=eh,e^c=deh,f^c=g,g^c=fg,g^d=gh,f^e=fh,g^f=gh35 conjugacy classes — view the full character table.
| derived subgroup of | #72242972 |
| maximal subgroup of | #32895⋊C3, #32897⋊C3, #11710347, … |
| maximal normal subgroup of | #32895⋊C3, #32897⋊C3, #11710347, … |
| maximal quotient of | #11710604, #62845⋊C3, #11710741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20124);