Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| derived | #997 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #1020 |
| commutator, hypocenter | #997 | C4 |
| Fitting | #585 | D3 |
| socle | C2 | #5936 |
| prime radical, cosocle | #5950 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C4 | |
165 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 14 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,8,28,21)(2,11,31,18)(3,20,19,14)(4,24,12,22)(5,10,25,32)(6,17,30,27)(7,16,26,23)(9,13,29,15);(4,15,5)(8,24,9)(11,28,12)(14,17,26)(16,19,18)(20,31,21)(23,25,30)(27,29,32)Polycyclic presentation:
2^2,3,2^5|a^2=b,b^2=h,f^2=h,g^2=h,c^a=c^2,d^a=e,e^a=d,f^a=gh,g^a=fh,d^c=e,e^c=de,f^c=fg,g^c=f,f^d=fh,g^e=gh,g^f=gh26 conjugacy classes — view the full character table.
| maximal subgroup of | #11711587, #11711588, #11711589, … |
| maximal normal subgroup of | #11711587, #11711588, #11711589, … |
| maximal quotient of | #11714226, #11714227, #11714230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18143);