Finite Group Databasebeta

← Prev (#16813401) Next (#16813403) →

C43×ΓL(1,𝔽8)⋊C4 #16813402

order3612
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4×C43 · Schur multiplier (H2) = C1

Aliases

direct productC43×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C43)
unambiguous semidirect productDic7⋊(C3×C43)

Splittings

derived, SylowC7 ⋊ C3×C4×C43
HallDic7×C43 ⋊ C3
HallΓL(1,𝔽8)×C43 ⋊ C4
HallC7×C43 ⋊ C3×C4
HallDic7 ⋊ C3×C43
HallΓL(1,𝔽8) ⋊ C4×C43

Subgroups

subgroupG/H
center, hypercenterC2×C43AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C43
FrattiniC2AGL(1,𝔽7)×C43
Fitting, socleC2×C7×C43C2×C3
prime radicalC2×ΓL(1,𝔽8)×C43C2
cosocleC2×C7C2×C3×C43
LCSG ▷ C7
UCSC1 ◁ C2×C43

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 1204

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)(45,49,48,46,47,50)(51,52,53,54);(44,45,48,49,47,50,46)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊C4×C43⋊C3, C3×C43×ΓL(1,𝔽8)⋊C4, C43×C3⋊(ΓL(1,𝔽8)⋊C4), …
maximal normal subgroup ofΓL(1,𝔽8)⋊C4×C43⋊C3, C3×C43×ΓL(1,𝔽8)⋊C4, C5×C43×ΓL(1,𝔽8)⋊C4, …
maximal quotient ofC43×Dic7⋊C9, C3×C43×ΓL(1,𝔽8)⋊C4, C43×C3⋊(ΓL(1,𝔽8)⋊C4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3612, 14);