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ΓL(1,𝔽8)⋊Dic19 #13097585

order1596
isoclinismisoclinic class of ΓL(1,𝔽8)⋊D19
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊Dic19

Splittings

derived, HallC7×C19 ⋊ C3×C4
HallDic133 ⋊ C3
HallC19×ΓL(1,𝔽8) ⋊ C4
HallΓL(1,𝔽8) ⋊ Dic19
SylowC19 ⋊ ΓL(1,𝔽8)⋊C4
SylowC7 ⋊ C3×Dic19

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)⋊D19
commutator, hypocenterC7×C19C3×C4
Fitting, socle, cosocleC2×C7×C19C2×C3
prime radicalC2×C19×ΓL(1,𝔽8)C2
LCSG ▷ C7×C19
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 532

Polycyclic presentation:

2,3,2,7,19|a^2=c,d^a=d^6,e^a=e^18,d^b=d^4

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊((C19⋊C3)⋊C4), C3×ΓL(1,𝔽8)⋊Dic19, C3⋊(ΓL(1,𝔽8)⋊Dic19), …
maximal normal subgroup ofΓL(1,𝔽8)⋊((C19⋊C3)⋊C4), C3×ΓL(1,𝔽8)⋊Dic19, C7×ΓL(1,𝔽8)⋊Dic19, …
maximal quotient of(C7⋊C9)⋊Dic19, C3×ΓL(1,𝔽8)⋊Dic19, C3⋊(ΓL(1,𝔽8)⋊Dic19), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1596, 8);