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C3⋊(C7⋊#10596817) #27131790

order12180
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C7⋊#10596817)
unambiguous semidirect productC7⋊(C3⋊#10596817)

Splittings

Fitting, socle, derived, HallC3×C5×C7×C29 ⋊ C4
HallC5×C7×C29 ⋊ Dic3
HallC3×C7×C29 ⋊ Dic5
HallC3×C5×C29 ⋊ Dic7
HallC7×C29 ⋊ Dic15
HallC5×C29 ⋊ Dic21
HallC3×C5×C7 ⋊ #1181
HallC3×C29 ⋊ Dic35
HallC5×C7 ⋊ C3⋊#1181
SylowC29 ⋊ Dic105
HallC3×C7 ⋊ #10596817
HallC3×C5 ⋊ C7⋊#1181
SylowC7 ⋊ C3⋊#10596817
SylowC5 ⋊ C3⋊(C7⋊#1181)
SylowC3 ⋊ C7⋊#10596817

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C29C4
LCSG ▷ C3×C5×C7×C29
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,5,7,29|a^2=b,c^a=c^2,d^a=d^4,e^a=e^6,f^a=f^17,f^b=f^28

Conjugacy classes & character table

843 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7⋊(C9⋊#10596817), C3⋊(ΓL(1,𝔽8)⋊#10596817), C3×C3⋊(C7⋊#10596817), …
maximal normal subgroup ofC3⋊(ΓL(1,𝔽8)⋊#10596817), C3×C3⋊(C7⋊#10596817)
maximal quotient ofC7⋊(C9⋊#10596817), C3×C3⋊(C7⋊#10596817), C3⋊(C3⋊(C7⋊#10596817))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12180, 125);