Finite Group Databasebeta

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C7×C3⋊(C7⋊#63120) #50304422

order38220
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×C3⋊(C7⋊#63120)
unambiguous semidirect productC3⋊(C7×C7⋊#63120)
unambiguous semidirect productC7⋊(C7×C3⋊#63120)

Splittings

Fitting, socle, HallC3×C5×C72×C13 ⋊ C4
derivedC3×C5×C7×C13 ⋊ C4×C7
HallC5×C72×C13 ⋊ Dic3
HallC3×C72×C13 ⋊ AGL(1,𝔽5)
HallC3×C5×C72 ⋊ Dic13
HallC72×C13 ⋊ ΓL(1,𝔽16)
HallC5×C72 ⋊ Dic39
HallC3×C5×C13 ⋊ C7×Dic7
HallC3×C72 ⋊ #63120
HallC5×C13 ⋊ C7×Dic21
SylowC72 ⋊ C3⋊#63120
HallC3×C13 ⋊ C7×C7⋊AGL(1,𝔽5)
HallC3×C5 ⋊ C7×Dic91
SylowC13 ⋊ C7×C3⋊(C7⋊AGL(1,𝔽5))
SylowC5 ⋊ C7×Dic273
SylowC3 ⋊ C7×C7⋊#63120

Subgroups

subgroupG/H
center, hypercenterC7C3⋊(C7⋊#63120)
commutator, hypocenterC3×C5×C7×C13C4×C7
FrattiniC1G
Fitting, socleC3×C5×C72×C13C4
prime radicalC3×C72×D5×C13C2
cosocleC3×C7×D5×C13C2×C7
LCSG ▷ C3×C5×C7×C13
UCSC1 ◁ C7

144 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 2730

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 1108);