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C3×C7×D61⋊C5 #27165455

order12810
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC3×C7×D61⋊C5
unambiguous semidirect product(C3×D61)⋊(C5×C7)
unambiguous semidirect product(C3×C7×D61)⋊C5

Splittings

Fitting, socle, HallC3×C7×C61 ⋊ C2×C5
derived, SylowC61 ⋊ C2×C3×C5×C7
HallC3×C7×C61⋊C5 ⋊ C2
HallC3×C7×D61 ⋊ C5
HallC7×C61⋊C5 ⋊ C2×C3
HallC3×C61⋊C5 ⋊ C2×C7
HallC7×D61 ⋊ C3×C5
HallC7×C61 ⋊ C2×C3×C5
HallC3×D61 ⋊ C5×C7
HallC61⋊C5 ⋊ C2×C3×C7
HallC3×C61 ⋊ C2×C5×C7
HallD61 ⋊ C3×C5×C7

Subgroups

subgroupG/H
center, hypercenterC3×C7D61⋊C5
commutator, cosocle, hypocenterC61C2×C3×C5×C7
FrattiniC1G
Fitting, socleC3×C7×C61C2×C5
LCSG ▷ C61
UCSC1 ◁ C3×C7

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 71
transitive degreeμt(G) = 1281

A minimal faithful permutation representation — 2 generators on 71 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC3×C7×#6923⋊C5, C3×C7⋊(#6923⋊C5), C3×D7×D61⋊C5, …
maximal normal subgroup ofC3×C7×#6923⋊C5, C3×C7⋊(#6923⋊C5), C3×D7×D61⋊C5, …
maximal quotient ofC3×C7×Dic61⋊C5, C2×C3×C7×D61⋊C5, C7×C9×D61⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 26);