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C7×A4×C17 #13083629

order1428
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C3×C7×C17 · Schur multiplier (H2) = C2

Aliases

direct productC7×A4×C17
unambiguous semidirect productC22⋊(C3×C7×C17)
unambiguous semidirect product(C22×C17)⋊(C3×C7)

Splittings

Fitting, socle, HallC22×C7×C17 ⋊ C3
derived, SylowC22 ⋊ C3×C7×C17
HallC22×C17 ⋊ C3×C7
HallC22×C7 ⋊ C3×C17

Subgroups

subgroupG/H
center, hypercenterC7×C17A4
commutator, cosocle, hypocenterC22C3×C7×C17
FrattiniC1G
Fitting, socleC22×C7×C17C3
LCSG ▷ C22
UCSC1 ◁ C7×C17

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 476

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofA4×C17×ΓL(1,𝔽8), C3×C7×A4×C17, C7×C11×A4×C17, …
maximal normal subgroup ofA4×C17×ΓL(1,𝔽8), C3×C7×A4×C17, C7×C11×A4×C17, …
maximal quotient ofC7×C17×C22⋊C9, C3×C7×A4×C17, C7×C11×A4×C17, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1428, 31);