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Dic7⋊SL(2,𝔽3) #10620802

order672
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productDic7⋊SL(2,𝔽3)
unambiguous semidirect product(C7⋊SL(2,𝔽3))⋊C4

Splittings

derivedC7×Q8 ⋊ C3×C4
HallQ8×Dic7 ⋊ C3
SylowC7 ⋊ C4×SL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC22D7⋊A4
commutator, hypocenterC7×Q8C3×C4
Fitting, cosocleC2×C7×Q8C2×C3
socleC22×C7C2×A4
prime radicalC2×C7⋊SL(2,𝔽3)C2
LCSG ▷ C7×Q8
UCSC1 ◁ C22

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Polycyclic presentation:

2,3,2^3,7,2|a^2=c,d^2=g,e^2=g,f^a=f^6,d^b=e,e^b=deg,f^b=f^4,e^d=eg

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#66665, #66665⋊C4, Dic14⋊SL(2,𝔽3), …
maximal normal subgroup ofC4×#66665, #66665⋊C4, Dic14⋊SL(2,𝔽3), …
maximal quotient of(C7⋊C8)⋊SL(2,𝔽3), C2×Dic7⋊SL(2,𝔽3), #72217686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 384);