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C2×D6325 #38760704

order25300
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD12650
direct productC2×D6325
unambiguous semidirect product(C2×C11)⋊D575
unambiguous semidirect product(C2×C23)⋊D275
unambiguous semidirect product(C2×C25)⋊D253

Splittings

FittingC2×C11×C23×C25 ⋊ C2
derived, HallC11×C23×C25 ⋊ C22
HallC23×C25 ⋊ C2×D11
HallC11×C25 ⋊ C2×D23
HallC11×C23 ⋊ C2×D25
SylowC25 ⋊ C2×D253
SylowC23 ⋊ C2×D275
SylowC11 ⋊ C2×D575

Subgroups

subgroupG/H
center, hypercenterC2D6325
commutator, cosocle, hypocenterC11×C23×C25C22
FrattiniC5C2×D1265
FittingC2×C11×C23×C25C2
socleC2×C5×C11×C23D5
LCSG ▷ C11×C23×C25
UCSC1 ◁ C2

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 61
transitive degreeμt(G) = 12650

A minimal faithful permutation representation — 2 generators on 61 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);(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(29,38)(30,37)(31,36)(32,35)(33,34)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Conjugacy classes & character table

6328 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD25300
maximal normal subgroup ofD25300
maximal quotient ofDic12650, D25300

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25300, 41);