Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4621 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #376 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #857 |
| commutator, cosocle, hypocenter | #95⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | #376 | D3 |
| prime radical | #4622 | C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C4 | |
113 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 113 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) = 28 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,12)(3,8,6)(4,9,11)(5,7,10)(13,27,22,20,23,28,17,18,24,14,15,26)(16,19,21,25);(2,6)(7,11)(8,10)(9,12)(15,21)(16,23)(18,25)(19,27)(22,24)(26,28)Polycyclic presentation:
2^2,3,2^5|b^2=h,d^2=f,e^2=fg,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=eh30 conjugacy classes — view the full character table.
| maximal subgroup of | #11711007, #11711008, #11711092, … |
| maximal normal subgroup of | #11711007, #11711008, #11711092, … |
| maximal quotient of | #11710978, C4×#4622, C4×#4621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5563);