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C2×D6405 #38785239

order25620
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD12810
direct productC2×D6405
unambiguous semidirect product(C2×C3)⋊D2135
unambiguous semidirect product(C2×C5)⋊D1281
unambiguous semidirect product(C2×C7)⋊D915
unambiguous semidirect product(C2×C61)⋊D105

Splittings

Fitting, socleC2×C3×C5×C7×C61 ⋊ C2
derived, HallC3×C5×C7×C61 ⋊ C22
HallC5×C7×C61 ⋊ C2×D3
HallC3×C7×C61 ⋊ C2×D5
HallC3×C5×C61 ⋊ C2×D7
HallC7×C61 ⋊ C2×D15
HallC5×C61 ⋊ C2×D21
HallC3×C61 ⋊ C2×D35
HallC3×C5×C7 ⋊ C2×D61
SylowC61 ⋊ C2×D105
HallC5×C7 ⋊ C2×D183
HallC3×C7 ⋊ C2×D305
HallC3×C5 ⋊ C2×D427
SylowC7 ⋊ C2×D915
SylowC5 ⋊ C2×D1281
SylowC3 ⋊ C2×D2135

Subgroups

subgroupG/H
center, hypercenterC2D6405
commutator, cosocle, hypocenterC3×C5×C7×C61C22
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C61C2
LCSG ▷ C3×C5×C7×C61
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 78
transitive degreeμt(G) = 12810

A minimal faithful permutation representation — 2 generators on 78 points:

(1,2)(3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78);(4,5)(7,10)(8,9)(12,17)(13,16)(14,15)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Conjugacy classes & character table

6408 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD25620
maximal normal subgroup ofD25620
maximal quotient ofDic12810, D25620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25620, 312);