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D3×C7×C61⋊C5 #27165458

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

Aliases

direct productD3×C7×C61⋊C5
unambiguous semidirect productC3⋊(C2×C7×C61⋊C5)
unambiguous semidirect product(D3×C61)⋊(C5×C7)
unambiguous semidirect product(D3×C7×C61)⋊C5

Splittings

Fitting, socle, HallC3×C7×C61 ⋊ C2×C5
derived, HallC3×C61 ⋊ C2×C5×C7
HallC3×C7×C61⋊C5 ⋊ C2
HallD3×C7×C61 ⋊ C5
HallC3×C61⋊C5 ⋊ C2×C7
HallC7×C61 ⋊ C5×D3
HallD3×C61 ⋊ C5×C7
SylowC61 ⋊ C5×D3×C7
HallC3×C7 ⋊ C2×C61⋊C5
SylowC3 ⋊ C2×C7×C61⋊C5

Subgroups

subgroupG/H
center, hypercenterC7D3×C61⋊C5
commutator, cosocle, hypocenterC3×C61C2×C5×C7
FrattiniC1G
Fitting, socleC3×C7×C61C2×C5
LCSG ▷ C3×C61
UCSC1 ◁ 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofD3×C7×D61⋊C5, D3×C7⋊(D61⋊C5), D3×D7×C61⋊C5, …
maximal normal subgroup ofD3×C7×D61⋊C5, D3×C7⋊(D61⋊C5), D3×D7×C61⋊C5, …
maximal quotient ofC7×Dic3×C61⋊C5, C2×D3×C7×C61⋊C5, C7×D9×C61⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 29);