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#63308⋊C3 #11719720

order840
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#63308⋊C3

Splittings

Hall#63308 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ Q8
HallC5×C7 ⋊ C3×Q8
HallΓL(1,𝔽8) ⋊ Dic10
SylowC7 ⋊ C3×Dic10
SylowC5 ⋊ ΓL(1,𝔽8)⋊Q8

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D5×AGL(1,𝔽7)
commutator, Fitting, socle, cosocleC2×C5×C7C22×C3
prime radicalC2×C5×ΓL(1,𝔽8)C22
hypocenterC5×C7C3×Q8
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Polycyclic presentation:

2^2,3,2,5,7|a^2=d,b^2=d,b^a=bd,f^a=f^6,e^b=e^4,f^c=f^4

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic10×AGL(1,𝔽7), D5×ΓL(1,𝔽8)⋊Q8, (C5⋊AGL(1,𝔽7))⋊Q8, …
maximal normal subgroup ofDic10×AGL(1,𝔽7), D5×ΓL(1,𝔽8)⋊Q8, (C5⋊AGL(1,𝔽7))⋊Q8, …
maximal quotient of(ΓL(1,𝔽8)⋊C4)⋊Dic5, #10587858⋊C3, #10587859⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 22);