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 | C22×Q8 ⋊ D3 |
| derived | C22⋊SL(2,𝔽3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C2 |
| Fitting | C22×Q8 | D3 |
| socle | C23 | S4 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,4,3)(5,6,11,7,8,10,9,12);(2,3,4)(6,9,12)(7,10,11)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,f^e=fg15 conjugacy classes — view the full character table.
| maximal subgroup of | #72999, #73000, #73002, … |
| maximal normal subgroup of | #72999, #73000, #73002, … |
| maximal quotient of | #85548, #85551, #85552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1490);