Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | H(7,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C24 |
| Frattini | C4 | C23×S4 |
| Fitting | H(7,𝔽2) | D3 |
| hypercenter | 2−1+4 | S4 |
| hypocenter | SL(2,𝔽3) | C22×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
1732 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 1732 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^5,3,2^3|d^2=i,e^2=i,g^2=i,h^2=i,d^a=de,e^a=ei,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,e^d=ei,g^f=h,h^f=ghi,h^g=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72224794, #72226022, C2×#11714723, … |
| maximal normal subgroup of | #72226022, C2×#11714723, #72226219, … |
| maximal quotient of | #72226116, C2×#11714723, D4×#5429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089104);