Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C22 |
| Frattini | C23 | C2×S4 |
| Fitting | B(4,𝔽2) | D3 |
| prime radical | B(4,𝔽2)⋊C3 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 | |
127 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,3,15)(2,6,16)(4,11)(5,9,10)(7,14)(8,12,13);(1,2,5,8)(3,7,9,14)(4,6,11,12)(10,13,15,16)Polycyclic presentation:
2^2,3,2^5|a^2=h,c^a=c^2,d^a=e,e^a=d,f^a=fg,d^b=dfgh,e^b=efh,d^c=e,e^c=de,f^c=g,g^c=fg,f^d=fh,g^d=gh,g^e=gh20 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73025, #11714162, #11714171, … |
| maximal normal subgroup of | C2×#73025, #11714162, #11714171, … |
| maximal quotient of | #11711402, #11711408, #11711420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5601);