Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | Aut(G) = #73026 (order 384) · Inn(G) = #5934 · Out(G) = C2 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | H(3,𝔽4) ⋊ D3 |
| derived | #5465 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5934 |
| commutator, cosocle, hypocenter | #5465 | C2 |
| Frattini, hypercenter | C22 | #1020 |
| Fitting | H(3,𝔽4) | D3 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C2 ◁ C22 | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,6,15,14,5,13,9,7)(2,3,16,4,8,10,12,11);(3,4,9)(6,7,12)(10,11,15)(13,14,16)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,e^c=eg,f^c=fh,e^d=eh,f^d=fgh16 conjugacy classes — view the full character table.
| maximal subgroup of | #11711671, #11711672, C2×#85557, … |
| maximal normal subgroup of | #11711671, #11711672, C2×#85557, … |
| maximal quotient of | #11714249, #11714251, #11714257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18133);