Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C23×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C22 |
| Fitting | C23×Q8 | D3 |
| socle | C23 | C2×S4 |
| hypercenter | C22 | #1020 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
267 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 17 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 — 2 generators on 20 points:
(2,3,4)(5,14,8)(6,20,12,10,13,16)(9,17,18)(11,19);(1,4,2,3)(5,6)(7,13)(8,16)(9,10)(11,14)(12,18)(15,20)(17,19)Polycyclic presentation:
2^2,3,2^5|f^2=h,g^2=h,b^a=bh,c^a=c^2,e^a=de,f^a=fg,g^a=gh,d^c=e,e^c=de,f^c=g,g^c=fgh,g^f=gh24 conjugacy classes — view the full character table.
| maximal subgroup of | #11711937, #11711938, #11711946, … |
| maximal normal subgroup of | #11711937, #11711938, #11711946, … |
| maximal quotient of | #11715583, #11715586, #11715589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20083);