Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×#1020⋊C2 (order 384) · Inn(G) = #1020 · Out(G) = C22 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | H(5,𝔽2) ⋊ D3 |
| derived | #997 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | #1020 |
| commutator, cosocle, hypocenter | #997 | C2 |
| Fitting | H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,6,8,3,4,2,5,7);(2,5,7)(3,6,8)Polycyclic presentation:
2,3,2^5|e^2=g,f^2=g,b^a=b^2,c^a=d,d^a=c,e^a=f,f^a=e,c^b=d,d^b=cd,e^b=ef,f^b=eg,e^c=eg,f^d=fg,f^e=fg13 conjugacy classes — view the full character table.
| maximal subgroup of | #73026, #73028, #73031, … |
| maximal normal subgroup of | #73026, #73028, #73031, … |
| maximal quotient of | #85550, #85551, #85553, … |
| holomorph of | Q8 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1494);