Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | GL(2,ℤ/4) |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C2×#67 |
| hypercenter | #41 | S4 |
| hypocenter | A4 | #135 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
309 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4)(6,11)(8,16)(10,17)(12,13)(14,20)(18,19);(5,7,9,15)(6,10,13,19)(8,14,16,20)(11,17,12,18);(2,3)(5,6)(7,10)(8,11)(9,13)(12,16)(14,17)(15,19)(18,20)Polycyclic presentation:
2^5,3,2^2|b^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^2,g^a=h,h^a=g,d^c=de,g^f=h,h^f=gh38 conjugacy classes — view the full character table.
| maximal subgroup of | #11712821, #11712823, #11713080, … |
| maximal normal subgroup of | #11712821, #11712823, #11713080, … |
| maximal quotient of | #11711962, #11711967, #11711972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18090);