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C5×C7×C3⋊Dic3 #11929504

order1260
isoclinismisoclinic class of C3⋊D3
automorphism?
homologyAbelianization (H1) = C4×C5×C7 · Schur multiplier (H2) = C3

Aliases

direct productC5×C7×C3⋊Dic3
unambiguous semidirect productC3⋊(C5×C7×Dic3)

Splittings

derived, SylowC32 ⋊ C4×C5×C7
HallC32×C5×C7 ⋊ C4
HallC32×C7 ⋊ C4×C5
HallC32×C5 ⋊ C4×C7

Subgroups

subgroupG/H
center, hypercenterC2×C5×C7C3⋊D3
commutator, hypocenterC32C4×C5×C7
FrattiniC2C5×C7×C3⋊D3
Fitting, socle, prime radicalC2×C32×C5×C7C2
cosocleC2×C32C2×C5×C7
LCSG ▷ C32
UCSC1 ◁ C2×C5×C7

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 1260

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14,15,16)(18,19)(21,22);(20,21,22);(17,18,19)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC7×C3⋊(C3⋊(Dic11⋊C5)), C7×C3⋊Dic3×C11⋊C5, C5×C7×C11×C3⋊Dic3, …
maximal normal subgroup ofC5×C7×C11×C3⋊Dic3, C5×C7×C13×C3⋊Dic3, C5×C7×C17×C3⋊Dic3, …
maximal quotient ofC7×C3⋊(C3⋊(Dic11⋊C5)), C7×C3⋊Dic3×C11⋊C5, C5×C7×C11×C3⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 56);