Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67993 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #4622 |
| commutator, prime radical, cosocle, hypocenter | #338⋊C3 | C2 |
| Frattini | C22×C4 | S4 |
| Fitting | #338 | D3 |
| hypercenter | C4 | #857 |
| LCS | G ▷ #338⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,24,13,22,18,25,28,11,6,16,30,8,5,7,17,27)(2,31,23,4,21,3,29,20,10,14,32,12,9,15,26,19);(3,12,28)(4,17,15)(7,22,29)(8,26,25)(11,23,24)(13,14,19)(16,27,32)(20,30,31)Polycyclic presentation:
2,3,2^6|a^2=h,c^2=e,d^2=ef,e^2=h,g^2=h,b^a=b^2,c^a=de,d^a=cefh,e^a=ef,g^a=gh,c^b=degh,d^b=cdg,e^b=efh,f^b=eg,d^c=dg,f^c=fh,e^d=eh,f^d=fh18 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67993, #11711358, #11711367, … |
| maximal normal subgroup of | C2×#67993, #11711358, #11711367, … |
| maximal quotient of | #11710957, C2×#67993, #11919921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 569);