Finite Group Databasebeta

← Prev (#13097583) Next (#13097585) →

C19×ΓL(1,𝔽8)⋊C4 #13097584

order1596
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4×C19 · Schur multiplier (H2) = C1

Aliases

direct productC19×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C19)
unambiguous semidirect productDic7⋊(C3×C19)

Splittings

derived, SylowC7 ⋊ C3×C4×C19
HallC19×Dic7 ⋊ C3
HallC19×ΓL(1,𝔽8) ⋊ C4
HallC7×C19 ⋊ C3×C4
HallDic7 ⋊ C3×C19
HallΓL(1,𝔽8) ⋊ C4×C19

Subgroups

subgroupG/H
center, hypercenterC2×C19AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C19
FrattiniC2C19×AGL(1,𝔽7)
Fitting, socleC2×C7×C19C2×C3
prime radicalC2×C19×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C3×C19
LCSG ▷ C7
UCSC1 ◁ C2×C19

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC19⋊C3×ΓL(1,𝔽8)⋊C4, C3×C19×ΓL(1,𝔽8)⋊C4, C19×C3⋊(ΓL(1,𝔽8)⋊C4), …
maximal normal subgroup ofC19⋊C3×ΓL(1,𝔽8)⋊C4, C3×C19×ΓL(1,𝔽8)⋊C4, C7×C19×ΓL(1,𝔽8)⋊C4, …
maximal quotient ofC19×Dic7⋊C9, C3×C19×ΓL(1,𝔽8)⋊C4, C19×C3⋊(ΓL(1,𝔽8)⋊C4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1596, 7);