Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #1020 |
| automorphism | Aut(G) = C2×(#1020⋊C3)⋊C2 (order 1152) · Inn(G) = #1020 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C22⋊A4 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #1020 |
| commutator, hypocenter | C22⋊A4 | C4 |
| Fitting, socle | C25 | D3 |
| prime radical, cosocle | C2×C22⋊A4 | C2 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,5)(4,12)(6,11)(7,8)(9,10);(1,4,9)(2,6,8)(3,7,11)(5,10,12)Polycyclic presentation:
2^2,3,2^4|a^2=b,c^a=c^2,d^a=e,e^a=d,f^a=g,g^a=f,d^c=e,e^c=de,f^c=fg,g^c=f20 conjugacy classes — view the full character table.
| maximal subgroup of | #72989, #1020⋊C4, #73080, … |
| maximal normal subgroup of | #72989, #1020⋊C4, #73080, … |
| maximal quotient of | #85561, #85562, #85563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1495);