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D87⋊D37 #27166173

order12876
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD87⋊D37
unambiguous semidirect productD111⋊D29
unambiguous semidirect productD1073⋊D3

Splittings

Fitting, socle, derived, HallC3×C29×C37 ⋊ C22
HallC29×C37 ⋊ C2×D3
HallC3×C37 ⋊ C2×D29
HallC3×C29 ⋊ C2×D37
SylowC37 ⋊ D3×D29
SylowC29 ⋊ D3×D37
SylowC3 ⋊ D29×D37

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C29×C37C22
LCSG ▷ C3×C29×C37
UCSC1

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 69
transitive degreeμt(G) = 3219

A minimal faithful permutation representation — 2 generators on 69 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(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)(68,69);(2,3)(4,5)(6,7)(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)(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)

Polycyclic presentation:

2^2,3,29,37|d^a=d^28,e^a=e^36,c^b=c^2,d^b=d^28

Conjugacy classes & character table

858 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD261⋊D37, D87⋊(D37⋊C3), C3×D87⋊D37, …
maximal normal subgroup ofD87⋊(D37⋊C3), C3×D87⋊D37
maximal quotient ofD261⋊D37, C3×D87⋊D37, D111⋊D87, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 42);