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ΓL(1,𝔽8)⋊Dic43 #16813403

order3612
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊Dic43

Splittings

derived, HallC7×C43 ⋊ C3×C4
HallDic301 ⋊ C3
HallΓL(1,𝔽8)×C43 ⋊ C4
SylowC43 ⋊ ΓL(1,𝔽8)⋊C4
HallΓL(1,𝔽8) ⋊ Dic43
SylowC7 ⋊ C3×Dic43

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)⋊D43
commutator, hypocenterC7×C43C3×C4
Fitting, socle, cosocleC2×C7×C43C2×C3
prime radicalC2×ΓL(1,𝔽8)×C43C2
LCSG ▷ C7×C43
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 1204

Polycyclic presentation:

2,3,7,43,2|a^2=e,c^a=c^6,d^a=d^42,c^b=c^4

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊((C43⋊C3)⋊C4), C3×ΓL(1,𝔽8)⋊Dic43, C3⋊(ΓL(1,𝔽8)⋊Dic43), …
maximal normal subgroup ofΓL(1,𝔽8)⋊((C43⋊C3)⋊C4), C3×ΓL(1,𝔽8)⋊Dic43, C5×ΓL(1,𝔽8)⋊Dic43, …
maximal quotient of(C7⋊C9)⋊Dic43, C3×ΓL(1,𝔽8)⋊Dic43, C3⋊(ΓL(1,𝔽8)⋊Dic43), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3612, 15);