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C3×C19⋊AGL(1,𝔽5) #11765893

order1140
isoclinismisoclinic class of C19⋊AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×C19⋊AGL(1,𝔽5)

Splittings

Fitting, socle, HallC3×C5×C19 ⋊ C4
derived, HallC5×C19 ⋊ C3×C4
HallC3×C19 ⋊ AGL(1,𝔽5)
SylowC19 ⋊ C3×AGL(1,𝔽5)
HallC3×C5 ⋊ Dic19
SylowC5 ⋊ C3×Dic19

Subgroups

subgroupG/H
center, hypercenterC3C19⋊AGL(1,𝔽5)
commutator, hypocenterC5×C19C3×C4
FrattiniC1G
Fitting, socleC3×C5×C19C4
prime radicalC3×D5×C19C2
cosocleD5×C19C2×C3
LCSG ▷ C5×C19
UCSC1 ◁ C3

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 285

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(24,25,26,27);(4,5,7,9,11,13,15,17,19,21,22,20,18,16,14,12,10,8,6)(23,24,27,25,26)

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4294⋊AGL(1,𝔽5), C9×C19⋊AGL(1,𝔽5), C3×(C19⋊C3)⋊AGL(1,𝔽5), …
maximal normal subgroup of#4294⋊AGL(1,𝔽5), C9×C19⋊AGL(1,𝔽5), C3×(C19⋊C3)⋊AGL(1,𝔽5), …
maximal quotient ofC9×C19⋊AGL(1,𝔽5), C32×C19⋊AGL(1,𝔽5), C3×C3⋊(C19⋊AGL(1,𝔽5)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1140, 21);