Finite Group Databasebeta

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C2×C3×C5⋊#11759816 #44478164

order30420
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3×C13

Aliases

direct productC2×C3×C5⋊#11759816
unambiguous semidirect product(C2×C5)⋊(C3×#11759816)

Splittings

Fitting, socleC2×C3×C5×C132 ⋊ C2×C3
derived, HallC5×C132 ⋊ C22×C32
HallC3×C5×#92789 ⋊ C22
HallC2×C5⋊(C13⋊D13) ⋊ C32
HallC3×#92789 ⋊ C2×D5
SylowC132 ⋊ C2×C32×D5
SylowC5 ⋊ C2×C3×#11759816

Subgroups

subgroupG/H
center, hypercenterC2×C3C5⋊#11759816
commutator, cosocle, hypocenterC5×C132C22×C32
FrattiniC1G
Fitting, socleC2×C3×C5×C132C2×C3
LCSG ▷ C5×C132
UCSC1 ◁ C2×C3

320 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 1170

A minimal faithful permutation representation — 2 generators on 36 points:

(7,8)(9,10)(12,16,17,13,14,18)(15,22,21,19,20,23)(25,29,30,26,27,31)(28,34,35,32,33,36);(1,2)(3,4,5)(6,7,9,10,8)(11,12,15,14,18,21,20,22,23,17,16,19,13)(24,25,28,30,29,33,35,36,34,27,31,32,26)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 753);