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Cayley diagram

C24×C4×D3 #87553

order384
isoclinismisoclinic class of D3
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C215

Aliases

direct productC24×C4×D3
unambiguous semidirect productC3⋊(C25×C4)
unambiguous semidirect product(C2×C3)⋊(C24×C4)

Splittings

FittingC24×C3×C4 ⋊ C2
derived, SylowC3 ⋊ C25×C4

Subgroups

subgroupG/H
center, hypercenterC24×C4D3
commutator, hypocenterC3C25×C4
FrattiniC2C25×D3
FittingC24×C3×C4C2
socleC25×C3C22
prime radicalC25×D3C2
cosocleC2×C3C26
LCSG ▷ C3
UCSC1 ◁ C24×C4

10552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 6 generators on 15 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×C22⋊#95, C2×(C2×D3)⋊#95, #10716392, …
maximal normal subgroup ofD3×C22⋊#95, C2×(C2×D3)⋊#95, #10716392, …
maximal quotient ofC23×C42×D3, C24×C4×Dic3, C23×D3×C22⋊C4, …
automorphism group ofDic3⋊C16, (C2×D3)⋊C16, C5×Dic3⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20129);