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C5×D5×D43⋊C3 #27166403

order12900
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D5×D43⋊C3
unambiguous semidirect productD43⋊(C3×C5×D5)
unambiguous semidirect product(D5×D43)⋊(C3×C5)
unambiguous semidirect product(C5×D43⋊C3)⋊(C2×C5)
unambiguous semidirect product(D5×C43⋊C3)⋊(C2×C5)
unambiguous semidirect product(C5⋊(D43⋊C3))⋊(C2×C5)
unambiguous semidirect product(C5×D5×D43)⋊C3

Splittings

Fitting, socle, HallC52×C43 ⋊ C22×C3
derivedC5×C43 ⋊ C22×C3×C5
HallC5×D5×D43 ⋊ C3
HallC52×C43⋊C3 ⋊ C22
HallC43⋊C3 ⋊ C2×C5×D5
SylowC43 ⋊ C2×C3×C5×D5
SylowC52 ⋊ C2×D43⋊C3

Subgroups

subgroupG/H
center, hypercenterC5D5×D43⋊C3
commutator, cosocle, hypocenterC5×C43C22×C3×C5
FrattiniC1G
Fitting, socleC52×C43C22×C3
LCSG ▷ C5×C43
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 430

A minimal faithful permutation representation — 2 generators on 53 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(44,45,46,47,48)(49,50,52,53,51);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(50,51)(52,53)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 72);