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Cayley diagram

C2×Dic3×M16 #83745

order384
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×Dic3×M16

Splittings

SylowC3 ⋊ C2×C4×M16

Subgroups

subgroupG/H
centerC22×C4C22×D3
commutatorC2×C3C22×C42
FrattiniC2×C4C23×D3
FittingC22×C3×M16C2
socleC23×C3C22×C4
prime radicalC24×C3C2×C4
cosocleC2×C3×C4C24
hypercenterC22×M16D3
hypocenterC3C2×C4×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×M16

656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×H(3,ℤ/4), Dic3×#338, Dic3×#339, …
maximal normal subgroup ofDic3×H(3,ℤ/4), Dic3×#338, Dic3×#339, …
maximal quotient ofC2×M16×C3⋊C8, C2×#70739, C2×#69847, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16321);