Finite Group Databasebeta

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Cayley diagram

C2×C4×D21 #66726

order336
isoclinismisoclinic class of D21
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4×D21

Splittings

FittingC2×C3×C4×C7 ⋊ C2
derived, HallC3×C7 ⋊ C22×C4
SylowC7 ⋊ C2×C4×D3
SylowC3 ⋊ C2×C4×D7

Subgroups

subgroupG/H
center, hypercenterC2×C4D21
commutator, hypocenterC3×C7C22×C4
FrattiniC2C22×D21
FittingC2×C3×C4×C7C2
socleC22×C3×C7C22
prime radicalC22×D21C2
cosocleC2×C3×C7C23
LCSG ▷ C3×C7
UCSC1 ◁ C2×C4

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620492, #10620659, C2×D21⋊C8, …
maximal normal subgroup of#10620492, #10620659, C2×D21⋊C8, …
maximal quotient ofC4×Dic42, C42×D21, #10621355, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 195);