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C3×C5⋊(C7⋊D77) #30633583

order16170
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C7

Aliases

direct productC3×C5⋊(C7⋊D77)
unambiguous semidirect productC5⋊(C3×C7⋊D77)
unambiguous semidirect productC11⋊(C3×C5⋊(C7⋊D7))

Splittings

Fitting, socle, HallC3×C5×C72×C11 ⋊ C2
derived, HallC5×C72×C11 ⋊ C2×C3
HallC3×C72×C11 ⋊ D5
HallC3×C5×C72 ⋊ D11
HallC72×C11 ⋊ C3×D5
HallC5×C72 ⋊ C3×D11
HallC3×C5×C11 ⋊ C7⋊D7
HallC3×C72 ⋊ D55
HallC5×C11 ⋊ C3×C7⋊D7
SylowC72 ⋊ C3×D55
HallC3×C11 ⋊ C5⋊(C7⋊D7)
HallC3×C5 ⋊ C7⋊D77
SylowC11 ⋊ C3×C5⋊(C7⋊D7)
SylowC5 ⋊ C3×C7⋊D77

Subgroups

subgroupG/H
center, hypercenterC3C5⋊(C7⋊D77)
commutator, cosocle, hypocenterC5×C72×C11C2×C3
FrattiniC1G
Fitting, socleC3×C5×C72×C11C2
LCSG ▷ C5×C72×C11
UCSC1 ◁ C3

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 8085

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(16,17)(18,19)(21,22)(23,24)(25,26)(28,29)(30,31)(32,33);(4,5,7,9,11,13,14,12,10,8,6)(15,16,18,19,17)(27,28,30,32,33,31,29);(20,21,23,25,26,24,22)

Conjugacy classes & character table

4047 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×D7×D385, C3×D35×D77, C3×D11×C5⋊(C7⋊D7), …
maximal normal subgroup ofC3×D7×D385, C3×D35×D77, C3×D11×C5⋊(C7⋊D7), …
maximal quotient ofC3×C7⋊Dic385, C2×C3×C5⋊(C7⋊D77)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 128);