Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C22 |
| Fitting | #585 | D3 |
| prime radical | #5950 | C2 |
| hypercenter | C4 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 ◁ C4 | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,3,6,14)(2,7,10,11)(4,13,15,27,5,32,16,29,22,12,18,26)(8,20,17,24,9,28,21,25,31,19,23,30);(1,17,16,23)(2,22,21,5)(3,28,27,20)(4,9,6,31)(7,26,24,29)(8,18,10,15)(11,32,19,13)(12,25,14,30)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,b^a=bh,c^a=c^2,e^a=de,f^a=fg,d^c=e,e^c=de,f^c=g,g^c=fg,f^d=fh,f^e=fh,g^e=gh23 conjugacy classes — view the full character table.
| maximal subgroup of | #11711640, #11711641, #11711645, … |
| maximal normal subgroup of | #11711640, #11711641, #11711645, … |
| maximal quotient of | #11715600, #11715601, #11715602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20093);