Finite Group Databasebeta

← Prev (#4583) Next (#4585) →

Cayley diagram

C3×H(3,ℤ/4) #4584

order192
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphismAut(G) = C2×#511819 (order 1024) · Inn(G) = C42 · Out(G) = C2×#120
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C2×C4

Aliases

direct productC3×H(3,ℤ/4)

Subgroups

subgroupG/H
centerC3×C4C42
commutatorC4C3×C42
Frattini, cosocleC22×C4C22×C3
socleC2×C3#95
prime radicalC23×C3D4
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C3×C4 ◁ G

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,19);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67481, #67482, #68696, …
maximal subgroup of#67481, #67482, C3×#1339, …
maximal normal subgroup of#67481, #67482, C3×#1339, …
maximal quotient ofC3×#1274, C3×#1275, C3×#1284, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 143);