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C2×C41⋊((C5⋊D5)⋊C3) #27134259

order12300
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C5

Aliases

direct productC2×C41⋊((C5⋊D5)⋊C3)
unambiguous semidirect product(C2×C41)⋊((C5⋊D5)⋊C3)
unambiguous semidirect product(C2×C5⋊D205)⋊C3

Splittings

Fitting, socleC2×C52×C41 ⋊ C2×C3
derived, HallC52×C41 ⋊ C22×C3
HallC2×C5⋊D205 ⋊ C3
HallC41×C52⋊C3 ⋊ C22
HallC52⋊C3 ⋊ C2×D41
SylowC41 ⋊ C2×(C5⋊D5)⋊C3
SylowC52 ⋊ C2×C3×D41

Subgroups

subgroupG/H
center, hypercenterC2C41⋊((C5⋊D5)⋊C3)
commutator, cosocle, hypocenterC52×C41C22×C3
FrattiniC1G
Fitting, socleC2×C52×C41C2×C3
LCSG ▷ C52×C41
UCSC1 ◁ C2

60 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

460 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×C41⋊((C5⋊D5)⋊C3), C2×C3×C41⋊((C5⋊D5)⋊C3), C2×C3⋊(C41⋊((C5⋊D5)⋊C3))
maximal normal subgroup ofC2×C3×C41⋊((C5⋊D5)⋊C3)
maximal quotient ofC2×C41⋊((C5⋊D5)⋊C9), D3×C41⋊((C5⋊D5)⋊C3), C2×C3×C41⋊((C5⋊D5)⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12300, 121);