Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #85559 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | H(3,𝔽4) ⋊ D3 |
| derived | #5464 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #5464 | C2 |
| Frattini, socle | C22 | #1020 |
| Fitting | H(3,𝔽4) | D3 |
| LCS | G ▷ #5464 | |
| UCS | C1 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,14,3,5,7,13,11)(4,8,6,16,12,9,15,10);(2,3,6)(4,5,12)(7,16,14)(8,10,15)(9,11,13)Polycyclic presentation:
2,3,2^6|b^a=b^2,c^a=d,d^a=c,e^a=f,f^a=e,g^a=gh,c^b=d,d^b=cd,e^b=ef,f^b=e,g^b=h,h^b=gh,e^c=eh,f^c=fgh,e^d=eg,f^d=fh| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 3 | 12 | 12 | 12 | 24 | 24 | 24 | 48 | 48 | 48 | 128 |
| order | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 3 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 |
| χ3 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | -1 |
| χ4 | 3 | 3 | 3 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 0 |
| χ5 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 |
| χ6 | 3 | 3 | -1 | 3 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 0 |
| χ7 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 0 |
| χ8 | 3 | 3 | -1 | -1 | 3 | 1 | -1 | 1 | -1 | -1 | 1 | 0 |
| χ9 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | 1 | 1 | -1 | 0 |
| χ10 | 6 | 6 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| χ11 | 12 | -4 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 |
| χ12 | 12 | -4 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 |
| maximal subgroup of | #11711670, C2×#85559, #11715753, … |
| maximal normal subgroup of | #11711670, C2×#85559, #11715753, … |
| maximal quotient of | #11714248, #11714250, #11714252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18135);