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C2×D2703 #24925221

order10812
automorphismAut(G) order 8995584
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD5406
direct productC2×D2703
unambiguous semidirect product(C2×C3)⋊D901
unambiguous semidirect product(C2×C17)⋊D159
unambiguous semidirect product(C2×C53)⋊D51

Splittings

Fitting, socleC2×C3×C17×C53 ⋊ C2
derived, HallC3×C17×C53 ⋊ C22
HallC17×C53 ⋊ C2×D3
HallC3×C53 ⋊ C2×D17
SylowC53 ⋊ C2×D51
HallC3×C17 ⋊ C2×D53
SylowC17 ⋊ C2×D159
SylowC3 ⋊ C2×D901

Subgroups

subgroupG/H
center, hypercenterC2D2703
commutator, cosocle, hypocenterC3×C17×C53C22
FrattiniC1G
Fitting, socleC2×C3×C17×C53C2
LCSG ▷ C3×C17×C53
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 75
transitive degreeμt(G) = 5406

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

Conjugacy classes & character table

2706 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×D8109, D3×D2703, C2×C3×D2703, …
maximal normal subgroup ofC2×C3×D2703, D10812
maximal quotient ofC2×D8109, D3×D2703, C2×C3×D2703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10812, 30);