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C19×C3⋊(ΓL(1,𝔽8)⋊C4) #18012180

order4788
isoclinismisoclinic class of C3⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4×C19 · Schur multiplier (H2) = C1

Aliases

direct productC19×C3⋊(ΓL(1,𝔽8)⋊C4)
unambiguous semidirect productC3⋊(C19×ΓL(1,𝔽8)⋊C4)
unambiguous semidirect productΓL(1,𝔽8)⋊(Dic3×C19)
unambiguous semidirect productDic21⋊(C3×C19)

Splittings

derivedC3×C7 ⋊ C3×C4×C19
HallC3×C19×ΓL(1,𝔽8) ⋊ C4
HallC7×C19 ⋊ C3×Dic3
HallC3×ΓL(1,𝔽8) ⋊ C4×C19
SylowC7 ⋊ C3×Dic3×C19

Subgroups

subgroupG/H
center, hypercenterC2×C19C3⋊AGL(1,𝔽7)
commutator, hypocenterC3×C7C3×C4×C19
FrattiniC2C19×C3⋊AGL(1,𝔽7)
Fitting, socleC2×C3×C7×C19C2×C3
prime radicalC2×C3×C19×ΓL(1,𝔽8)C2
cosocleC2×C3×C7C2×C3×C19
LCSG ▷ C3×C7
UCSC1 ◁ C2×C19

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 1596

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,24,25,22,23,26)(28,29)(30,31,32,33);(20,21,23,26,25,24,22)(27,28,29)

Conjugacy classes & character table

456 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×C19×C3⋊(ΓL(1,𝔽8)⋊C4), C19×C3⋊(ΓL(1,𝔽8)⋊Dic5), C19×C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)), …
maximal normal subgroup ofC5×C19×C3⋊(ΓL(1,𝔽8)⋊C4), C7×C19×C3⋊(ΓL(1,𝔽8)⋊C4)
maximal quotient ofC5×C19×C3⋊(ΓL(1,𝔽8)⋊C4), C19×C3⋊(ΓL(1,𝔽8)⋊Dic5), C19×C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 96);