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Cayley diagram

#1157⋊C3 #66655

order336
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphismAut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C23
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect product#1157⋊C3

Splittings

FittingC2×C4×C7 ⋊ C2×C3
Hall#1157 ⋊ C3
HallΓL(1,𝔽8) ⋊ #41
SylowC7 ⋊ C3×#41

Subgroups

subgroupG/H
centerC4C2×AGL(1,𝔽7)
commutator, socle, cosocleC2×C7C23×C3
FrattiniC2C22×AGL(1,𝔽7)
FittingC2×C4×C7C2×C3
hypercenterC2×C4AGL(1,𝔽7)
hypocenterC7C3×#41
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 15 points:

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

Polycyclic presentation:

2^3,3,2,7|b^2=e,c^a=ce,f^a=f^6,f^d=f^4

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6379⋊C3, #6399⋊C3, #6464⋊C3, …
maximal normal subgroup of#6379⋊C3, #6399⋊C3, #6464⋊C3, …
maximal quotient ofC4×ΓL(1,𝔽8)⋊Q8, #6433⋊C3, #6435⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 124);