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, socle, hypercenter | C22 | #1020 |
| commutator, hypocenter | #997 | C4 |
| Fitting | C2×H(5,𝔽2) | D3 |
| prime radical, cosocle | C2×#997 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 | |
199 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 199 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) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4)(5,6,10,7,8,11,12,9);(6,10,7)(9,11,12)Polycyclic presentation:
2^2,3,2^5|a^2=b,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 | #11711578, #11711579, #11711580, … |
| maximal normal subgroup of | #11711578, #11711579, #11711580, … |
| maximal quotient of | #11714226, #11714228, #11714229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18141);