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C7×D15×D57 #37561565

order23940
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×D15×D57

Splittings

Fitting, socle, HallC32×C5×C7×C19 ⋊ C22
derived, HallC32×C5×C19 ⋊ C22×C7
HallC32×C7×C19 ⋊ C2×D5
HallC5×C7×C19 ⋊ D32
HallC32×C5×C7 ⋊ C2×D19
HallC32×C19 ⋊ C2×C7×D5
HallC7×C19 ⋊ D3×D15
HallC5×C19 ⋊ D32×C7
HallC32×C7 ⋊ D5×D19
HallC32×C5 ⋊ C2×C7×D19
HallC5×C7 ⋊ D3×D57
SylowC19 ⋊ D3×C7×D15
SylowC32 ⋊ C7×D5×D19
SylowC5 ⋊ D3×C7×D57

Subgroups

subgroupG/H
center, hypercenterC7D15×D57
commutator, cosocle, hypocenterC32×C5×C19C22×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C19C22
LCSG ▷ C32×C5×C19
UCSC1 ◁ C7

176 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 3990

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

1890 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 697);