Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C22 |
| Fitting | C2×H(5,𝔽2) | D3 |
| hypercenter | C22 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 ◁ C22 | |
276 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 16 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:
(1,11,4)(2,16,8,6,9,12)(5,13,14)(7,15);(1,2)(3,16,10,9)(4,8)(5,6)(7,13,15,11)(12,14)Polycyclic presentation:
2^2,3,2^5|f^2=h,g^2=h,b^a=bh,c^a=c^2,e^a=de,f^a=fg,g^a=gh,d^c=e,e^c=de,f^c=g,g^c=fgh,f^d=fh,f^e=fh,g^e=gh,g^f=gh23 conjugacy classes — view the full character table.
| maximal subgroup of | #11714170, #11714171, #11714172, … |
| maximal normal subgroup of | #11714170, #11714171, #11714172, … |
| maximal quotient of | #11715583, #11715586, #11715591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20092);