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C5×C3⋊Dic609 #49141356

order36540
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊Dic609
unambiguous semidirect productC3⋊(C5×Dic609)
unambiguous semidirect productC7⋊(C5×C3⋊Dic87)

Splittings

derived, HallC32×C7×C29 ⋊ C4×C5
HallC32×C5×C7×C29 ⋊ C4
HallC32×C5×C29 ⋊ Dic7
HallC5×C7×C29 ⋊ C3⋊Dic3
HallC32×C5×C7 ⋊ Dic29
HallC32×C29 ⋊ C5×Dic7
HallC7×C29 ⋊ C5×C3⋊Dic3
HallC5×C29 ⋊ C3⋊Dic21
HallC32×C7 ⋊ C5×Dic29
HallC32×C5 ⋊ Dic203
HallC5×C7 ⋊ C3⋊Dic87
SylowC29 ⋊ C5×C3⋊Dic21
SylowC32 ⋊ C5×Dic203
SylowC7 ⋊ C5×C3⋊Dic87

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊D609
commutator, hypocenterC32×C7×C29C4×C5
FrattiniC2C5×C3⋊D609
Fitting, socle, prime radicalC2×C32×C5×C7×C29C2
cosocleC2×C32×C7×C29C2×C5
LCSG ▷ C32×C7×C29
UCSC1 ◁ C2×C5

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 36540

A minimal faithful permutation representation — 3 generators on 51 points:

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

Conjugacy classes & character table

9150 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 198);