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C2×D3111 #27159209

order12444
automorphismAut(G) order 11946240
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD6222
direct productC2×D3111
unambiguous semidirect product(C2×C3)⋊D1037
unambiguous semidirect product(C2×C17)⋊D183
unambiguous semidirect product(C2×C61)⋊D51

Splittings

Fitting, socleC2×C3×C17×C61 ⋊ C2
derived, HallC3×C17×C61 ⋊ C22
HallC17×C61 ⋊ C2×D3
HallC3×C61 ⋊ C2×D17
SylowC61 ⋊ C2×D51
HallC3×C17 ⋊ C2×D61
SylowC17 ⋊ C2×D183
SylowC3 ⋊ C2×D1037

Subgroups

subgroupG/H
center, hypercenterC2D3111
commutator, cosocle, hypocenterC3×C17×C61C22
FrattiniC1G
Fitting, socleC2×C3×C17×C61C2
LCSG ▷ C3×C17×C61
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 83
transitive degreeμt(G) = 6222

A minimal faithful permutation representation — 2 generators on 83 points:

(1,2)(3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(4,5)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Conjugacy classes & character table

3114 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×D9333, C2×C3⋊(C17⋊(D61⋊C3)), D3×D3111, …
maximal normal subgroup ofC2×C3⋊(C17⋊(D61⋊C3)), C2×C3×D3111, D12444
maximal quotient ofC2×D9333, D3×D3111, C2×C3×D3111, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12444, 46);