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C2×D6045 #37564013

order24180
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD12090
direct productC2×D6045
unambiguous semidirect product(C2×C3)⋊D2015
unambiguous semidirect product(C2×C5)⋊D1209
unambiguous semidirect product(C2×C13)⋊D465
unambiguous semidirect product(C2×C31)⋊D195

Splittings

Fitting, socleC2×C3×C5×C13×C31 ⋊ C2
derived, HallC3×C5×C13×C31 ⋊ C22
HallC5×C13×C31 ⋊ C2×D3
HallC3×C13×C31 ⋊ C2×D5
HallC3×C5×C31 ⋊ C2×D13
HallC13×C31 ⋊ C2×D15
HallC3×C5×C13 ⋊ C2×D31
HallC5×C31 ⋊ C2×D39
HallC3×C31 ⋊ C2×D65
HallC5×C13 ⋊ C2×D93
HallC3×C13 ⋊ C2×D155
SylowC31 ⋊ C2×D195
HallC3×C5 ⋊ C2×D403
SylowC13 ⋊ C2×D465
SylowC5 ⋊ C2×D1209
SylowC3 ⋊ C2×D2015

Subgroups

subgroupG/H
center, hypercenterC2D6045
commutator, cosocle, hypocenterC3×C5×C13×C31C22
FrattiniC1G
Fitting, socleC2×C3×C5×C13×C31C2
LCSG ▷ C3×C5×C13×C31
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 12090

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2)(3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54);(4,5)(7,10)(8,9)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Conjugacy classes & character table

6048 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD24180
maximal normal subgroup ofD24180
maximal quotient ofDic12090, D24180

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24180, 312);