Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×#120 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1020⋊C2 |
| commutator, cosocle | C2×C22⋊A4 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C2×#120 | D3 |
| hypocenter | C22⋊A4 | D4 |
| LCS | G ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 | |
247 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 247 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) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,6,8,5,7,11)(2,4,10,3,9,12);(2,5)(4,8)(6,10)(7,12)(9,11)Polycyclic presentation:
2^3,3,2^4|b^a=bc,d^a=d^2,e^a=f,f^a=e,g^a=gh,e^b=egh,f^b=fg,e^d=f,f^d=ef,g^d=h,h^d=gh22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73081, #11713151, #11713200, … |
| maximal normal subgroup of | C2×#73081, #11713151, #11713200, … |
| maximal quotient of | #11711237, #11711238, #11711241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5657);