Finite Group Databasebeta

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Cayley diagram

C2×D5×C3⋊D3 #67222

order360
isoclinismisoclinic class of D5×C3⋊D3
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23×C3

Aliases

direct productC2×D5×C3⋊D3

Splittings

Fitting, socleC2×C32×C5 ⋊ C22
derived, HallC32×C5 ⋊ C23
SylowC32 ⋊ C22×D5
SylowC5 ⋊ C22×C3⋊D3

Subgroups

subgroupG/H
center, hypercenterC2D5×C3⋊D3
commutator, cosocle, hypocenterC32×C5C23
FrattiniC1G
Fitting, socleC2×C32×C5C22
LCSG ▷ C32×C5
UCSC1 ◁ C2

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 90

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽5))⋊(C3⋊D3), #10623978, #10623985, …
maximal normal subgroup of(C2×AGL(1,𝔽5))⋊(C3⋊D3), #10623978, #10623985, …
maximal quotient ofD5×#634, #10624148, #10624149, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 152);