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Cayley diagram

D7×SL(2,𝔽3) #66663

order336
isoclinismisoclinic class of #66662
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD7×SL(2,𝔽3)

Splittings

Fitting, derivedC7×Q8 ⋊ C2×C3
HallQ8×D7 ⋊ C3
SylowC7 ⋊ C2×SL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2A4×D7
commutator, Fitting, cosocle, hypocenterC7×Q8C2×C3
socleC2×C7C2×A4
LCSG ▷ C7×Q8
UCSC1 ◁ C2

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

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

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×#226, #10621474, #10621475, …
maximal normal subgroup ofD7×#226, #10621474, #10621475, …
maximal quotient ofSL(2,𝔽3)×Dic7, C2×D7×SL(2,𝔽3), D7×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 132);