Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C2×#95 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#857 |
| commutator, cosocle, hypocenter | #95⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C2×#95 | D3 |
| hypercenter | C22 | #857 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C22 | |
137 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,11,22)(2,24,18,6,17,9)(3,20,4)(5,12,21)(7,23,16,15,19,8)(10,14,13);(1,2)(3,9)(4,8)(5,6)(7,10)(11,19)(12,23)(13,18)(14,16)(15,20)(17,21)(22,24)Polycyclic presentation:
2^2,3,2^5|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 | C2×#72985, #11711819, #11711820, … |
| maximal normal subgroup of | C2×#72985, #11711819, #11711820, … |
| maximal quotient of | #11710979, #11711005, #11711006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5561);