Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #376 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#857 |
| commutator, cosocle, hypocenter | #95⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | #376 | D3 |
| hypercenter | C4 | #857 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,3,23,7,11,19,6,13,24,2,8,22)(4,18,10,15,14,16,9,12,20,5,21,17);(2,7)(3,10)(4,5)(8,16)(9,15)(11,17)(12,22)(13,20)(14,23)(18,19)(21,24)Polycyclic presentation:
2^2,3,2^5|b^2=h,d^2=f,e^2=fg,b^a=bh,c^a=c^2,d^a=ef,e^a=dfg,f^a=fg,d^c=efh,e^c=deh,f^c=fg,g^c=fh,e^d=eh24 conjugacy classes — view the full character table.
| maximal subgroup of | #11711211, #11711212, #11711227, … |
| maximal normal subgroup of | #11711211, #11711212, #11711227, … |
| maximal quotient of | #11710980, #11711005, #11711165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5564);