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C5×D3×D407 #37567596

order24420
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D3×D407
unambiguous semidirect product(D3×C37)⋊(C5×D11)

Splittings

Fitting, socle, HallC3×C5×C11×C37 ⋊ C22
derived, HallC3×C11×C37 ⋊ C22×C5
HallC5×C11×C37 ⋊ C2×D3
HallC3×C5×C37 ⋊ C2×D11
HallC11×C37 ⋊ C2×C5×D3
HallC5×C37 ⋊ D3×D11
HallC3×C5×C11 ⋊ C2×D37
HallC3×C37 ⋊ C2×C5×D11
HallC5×C11 ⋊ D3×D37
SylowC37 ⋊ C5×D3×D11
HallC3×C11 ⋊ C2×C5×D37
HallC3×C5 ⋊ C2×D407
SylowC11 ⋊ C5×D3×D37
SylowC3 ⋊ C2×C5×D407

Subgroups

subgroupG/H
center, hypercenterC5D3×D407
commutator, cosocle, hypocenterC3×C11×C37C22×C5
FrattiniC1G
Fitting, socleC3×C5×C11×C37C22
LCSG ▷ C3×C11×C37
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 6105

A minimal faithful permutation representation — 2 generators on 56 points:

(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);(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)(38,39,41,43,45,47,48,46,44,42,40)(50,51)

Conjugacy classes & character table

3075 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24420, 117);