Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C22⋊C4 ⋊ D3 |
250 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 26 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:
(1,2,4)(5,7)(6,9,10,11)(8,12);(2,3)(5,6)(7,11)(8,10)(9,12)Polycyclic presentation:
2^5,3,2^2|b^2=c,b^a=bd,c^a=ce,f^a=f^2,g^a=h,h^a=g,d^b=de,g^f=h,h^f=gh32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73049, #11712753, S4×#99, … |
| maximal normal subgroup of | C2×#73049, #11712753, S4×#99, … |
| maximal quotient of | #11711146, #11711147, #11711152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5625);