Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #1020⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C22⋊A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1020⋊C2 |
| commutator, hypocenter | C22⋊A4 | C2×C4 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C2×#120 | D3 |
| prime radical | C2×#120⋊C3 | C2 |
| cosocle | C2×C22⋊A4 | C22 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 | |
215 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 20 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) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(5,9,7,10,8,6)(11,12);(1,2,3,4)(7,12)(9,10)Polycyclic presentation:
2^3,3,2^4|a^2=c,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=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | #5450⋊C4, #4625⋊C4, C2×#73080, … |
| maximal normal subgroup of | #5450⋊C4, #4625⋊C4, C2×#73080, … |
| maximal quotient of | #11711113, #11711115, #11711126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5656);