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C7×(D61⋊C3)⋊C5 #27165490

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

Aliases

direct productC7×(D61⋊C3)⋊C5
unambiguous semidirect product(D61⋊C3)⋊(C5×C7)
unambiguous semidirect product(D61⋊C5)⋊(C3×C7)
unambiguous semidirect product((C61⋊C3)⋊C5)⋊(C2×C7)
unambiguous semidirect product(C7×D61⋊C3)⋊C5

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC7(D61⋊C3)⋊C5
commutator, cosocle, hypocenterC61C2×C3×C5×C7
FrattiniC1G
Fitting, socleC7×C61C2×C3×C5
LCSG ▷ C61
UCSC1 ◁ C7

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 427

A minimal faithful permutation representation — 2 generators on 68 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×AGL(1,𝔽61), #19103988⋊C5, D7×(D61⋊C3)⋊C5, …
maximal normal subgroup ofC7×AGL(1,𝔽61), #19103988⋊C5, D7×(D61⋊C3)⋊C5, …
maximal quotient ofC7×(Dic61⋊C3)⋊C5, C2×C7×(D61⋊C3)⋊C5, C7×(D61⋊C5)⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 61);