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C2×D2737 #24926517

order10948
automorphismAut(G) order 11561088
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD5474
direct productC2×D2737
unambiguous semidirect product(C2×C7)⋊D391
unambiguous semidirect product(C2×C17)⋊D161
unambiguous semidirect product(C2×C23)⋊D119

Splittings

Fitting, socleC2×C7×C17×C23 ⋊ C2
derived, HallC7×C17×C23 ⋊ C22
HallC17×C23 ⋊ C2×D7
HallC7×C23 ⋊ C2×D17
HallC7×C17 ⋊ C2×D23
SylowC23 ⋊ C2×D119
SylowC17 ⋊ C2×D161
SylowC7 ⋊ C2×D391

Subgroups

subgroupG/H
center, hypercenterC2D2737
commutator, cosocle, hypocenterC7×C17×C23C22
FrattiniC1G
Fitting, socleC2×C7×C17×C23C2
LCSG ▷ C7×C17×C23
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 5474

A minimal faithful permutation representation — 2 generators on 49 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);(4,9)(5,8)(6,7)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)

Conjugacy classes & character table

2740 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C17⋊(ΓL(1,𝔽8)⋊D23), D3×D2737, C2×C3×D2737, …
maximal normal subgroup ofC2×C17⋊(ΓL(1,𝔽8)⋊D23), C2×C3×D2737, D10948
maximal quotient ofD3×D2737, C2×C3×D2737, C2×D8211, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10948, 21);