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C7×D15×C61 #27165451

order12810
isoclinismisoclinic class of D15
automorphism?
homologyAbelianization (H1) = C2×C7×C61 · Schur multiplier (H2) = C1

Aliases

direct productC7×D15×C61
unambiguous semidirect productC3⋊(C7×D5×C61)
unambiguous semidirect productC5⋊(D3×C7×C61)

Splittings

Fitting, socle, HallC3×C5×C7×C61 ⋊ C2
derived, HallC3×C5 ⋊ C2×C7×C61
HallC5×C7×C61 ⋊ D3
HallC3×C7×C61 ⋊ D5
HallC3×C5×C61 ⋊ C2×C7
HallC5×C61 ⋊ D3×C7
HallC3×C61 ⋊ C7×D5
HallC3×C5×C7 ⋊ C2×C61
HallC5×C7 ⋊ D3×C61
HallC3×C7 ⋊ D5×C61
SylowC5 ⋊ D3×C7×C61
SylowC3 ⋊ C7×D5×C61

Subgroups

subgroupG/H
center, hypercenterC7×C61D15
commutator, cosocle, hypocenterC3×C5C2×C7×C61
FrattiniC1G
Fitting, socleC3×C5×C7×C61C2
LCSG ▷ C3×C5
UCSC1 ◁ C7×C61

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 76
transitive degreeμt(G) = 6405

A minimal faithful permutation representation — 2 generators on 76 points:

(2,3)(4,5)(7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)(70,71,72,73,74,75,76);(1,2,4,5,3)(6,7,8)

Conjugacy classes & character table

3843 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×D15×D61, D7×D15×C61, D15×D427, …
maximal normal subgroup ofC7×D15×D61, D7×D15×C61, D15×D427, …
maximal quotient ofC7×Dic15×C61, C2×C7×D15×C61, C7×C61×D45, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 22);