Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | #1020 |
| commutator, prime radical, cosocle, hypocenter | #5465 | C2 |
| Fitting | H(3,𝔽4) | D3 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C22 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,6,3,4,7,8,5)(9,20,21,15,13,14,18,17)(10,11,24,22,16,19,23,12);(2,6,3)(5,7,8)(11,12,21)(14,15,24)(17,23,20)(18,19,22)Polycyclic presentation:
2,3,2^6|a^2=h,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=b^2,c^a=dh,d^a=ch,e^a=fgh,f^a=egh,c^b=d,d^b=cdh,e^b=ef,f^b=e,d^c=dh,e^c=eg,f^d=fg,f^e=fgh19 conjugacy classes — view the full character table.
| maximal subgroup of | #11711672, C2×#85556, #11715639, … |
| maximal normal subgroup of | #11711672, C2×#85556, #11715639, … |
| maximal quotient of | #11714221, #11714222, #11714235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18132);