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C3×D7⋊(C22⋊A4) #72244642

order2016
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C24×C3

Aliases

direct productC3×D7⋊(C22⋊A4)

Splittings

Fitting, socleC24×C3×C7 ⋊ C2×C3
derivedC24×C7 ⋊ C2×C32
HallC24×D7 ⋊ C32
SylowC7 ⋊ C2×C3×C22⋊A4

Subgroups

subgroupG/H
center, hypercenterC3D7⋊(C22⋊A4)
commutator, cosocle, hypocenterC24×C7C2×C32
FrattiniC1G
Fitting, socleC24×C3×C7C2×C3
LCSG ▷ C24×C7
UCSC1 ◁ C3

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 252

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2)(3,5)(6,7)(12,15)(13,14)(16,17,18);(2,4,5)(3,6,7)(9,11,10)(13,15,14);(8,11)(9,10)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×C22⋊A4, D3×D7⋊(C22⋊A4)
maximal normal subgroup ofC3×AGL(1,𝔽7)×C22⋊A4, D3×D7⋊(C22⋊A4)
maximal quotient ofC3×D7⋊#997

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(3,5)(6,7)(12,15)(13,14)(16,17,18), (2,4,5)(3,6,7)(9,11,10)(13,15,14), (8,11)(9,10));