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Cayley diagram

A4×Dic7 #66664

order336
isoclinismisoclinic class of A4×D7
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productA4×Dic7

Splittings

derivedC22×C7 ⋊ C3×C4
HallC22×Dic7 ⋊ C3
SylowC7 ⋊ C4×A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2A4×D7
commutator, hypocenterC22×C7C3×C4
Fitting, socle, cosocleC23×C7C2×C3
prime radicalC2×C7×A4C2
LCSG ▷ C22×C7
UCSC1 ◁ C2

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10621477, S4×Dic7, (C7⋊S4)⋊C4, …
maximal normal subgroup of#10621477, S4×Dic7, (C7⋊S4)⋊C4, …
maximal quotient ofA4×C7⋊C8, SL(2,𝔽3)×Dic7, #10620795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 133);