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C5×C7×D61⋊C3 #27165462

order12810
isoclinismisoclinic class of D61⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC5×C7×D61⋊C3
unambiguous semidirect product(C5×D61)⋊(C3×C7)

Splittings

Fitting, socle, HallC5×C7×C61 ⋊ C2×C3
derived, SylowC61 ⋊ C2×C3×C5×C7
HallC5×C7×C61⋊C3 ⋊ C2
HallC5×C7×D61 ⋊ C3
HallC7×C61⋊C3 ⋊ C2×C5
HallC5×C61⋊C3 ⋊ C2×C7
HallC7×D61 ⋊ C3×C5
HallC5×D61 ⋊ C3×C7
HallC7×C61 ⋊ C2×C3×C5
HallC5×C61 ⋊ C2×C3×C7
HallC61⋊C3 ⋊ C2×C5×C7
HallD61 ⋊ C3×C5×C7

Subgroups

subgroupG/H
center, hypercenterC5×C7D61⋊C3
commutator, cosocle, hypocenterC61C2×C3×C5×C7
FrattiniC1G
Fitting, socleC5×C7×C61C2×C3
LCSG ▷ C61
UCSC1 ◁ C5×C7

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 73
transitive degreeμt(G) = 2135

A minimal faithful permutation representation — 2 generators on 73 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC5×C7×#6923⋊C3, C5×#19103988, C5×D7×D61⋊C3, …
maximal normal subgroup ofC5×C7×#6923⋊C3, C5×#19103988, C5×D7×D61⋊C3, …
maximal quotient ofC5×C7×Dic61⋊C3, C2×C5×C7×D61⋊C3, C5×C7×D61⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 33);