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C2×D13175 #72289701

order52700
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD26350
direct productC2×D13175

Splittings

FittingC2×C17×C25×C31 ⋊ C2
derived, HallC17×C25×C31 ⋊ C22
HallC25×C31 ⋊ C2×D17
HallC17×C31 ⋊ C2×D25
HallC17×C25 ⋊ C2×D31
SylowC31 ⋊ C2×D425
SylowC25 ⋊ C2×D527
SylowC17 ⋊ C2×D775

Subgroups

subgroupG/H
center, hypercenterC2D13175
commutator, cosocle, hypocenterC17×C25×C31C22
FrattiniC5C2×D2635
FittingC2×C17×C25×C31C2
socleC2×C5×C17×C31D5
LCSG ▷ C17×C25×C31
UCSC1 ◁ C2

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 75
transitive degreeμt(G) = 26350

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

Conjugacy classes & character table

13178 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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