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Cayley diagram

D5×C3⋊Dic3 #67135

order360
isoclinismisoclinic class of D5×C3⋊D3
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productD5×C3⋊Dic3

Splittings

derived, HallC32×C5 ⋊ C2×C4
SylowC32 ⋊ C4×D5
SylowC5 ⋊ C2×C3⋊Dic3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D5×C3⋊D3
commutator, hypocenterC32×C5C2×C4
Fitting, socle, cosocleC2×C32×C5C22
prime radicalC2×C32×D5C2
LCSG ▷ C32×C5
UCSC1 ◁ C2

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C3⋊Dic3, #10623965, ΓL(1,𝔽16)⋊Dic3, …
maximal normal subgroup ofAGL(1,𝔽5)×C3⋊Dic3, #10623965, ΓL(1,𝔽16)⋊Dic3, …
maximal quotient ofD5×C3⋊(C3⋊C8), #10623754, Dic5×C3⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 65);