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C3×C5⋊(C7⋊D161) #46798976

order33810
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C7

Aliases

direct productC3×C5⋊(C7⋊D161)
unambiguous semidirect productC5⋊(C3×C7⋊D161)
unambiguous semidirect productC23⋊(C3×C5⋊(C7⋊D7))

Splittings

Fitting, socle, HallC3×C5×C72×C23 ⋊ C2
derived, HallC5×C72×C23 ⋊ C2×C3
HallC3×C72×C23 ⋊ D5
HallC72×C23 ⋊ C3×D5
HallC3×C5×C72 ⋊ D23
HallC3×C5×C23 ⋊ C7⋊D7
HallC5×C72 ⋊ C3×D23
HallC3×C72 ⋊ D115
HallC5×C23 ⋊ C3×C7⋊D7
HallC3×C23 ⋊ C5⋊(C7⋊D7)
SylowC72 ⋊ C3×D115
SylowC23 ⋊ C3×C5⋊(C7⋊D7)
HallC3×C5 ⋊ C7⋊D161
SylowC5 ⋊ C3×C7⋊D161

Subgroups

subgroupG/H
center, hypercenterC3C5⋊(C7⋊D161)
commutator, cosocle, hypocenterC5×C72×C23C2×C3
FrattiniC1G
Fitting, socleC3×C5×C72×C23C2
LCSG ▷ C5×C72×C23
UCSC1 ◁ C3

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 16905

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3)(5,6)(7,8)(10,11)(12,13)(14,15)(17,18)(19,20)(21,22)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45);(4,5,7,8,6)(16,17,19,21,22,20,18)(23,24,26,28,30,32,34,36,38,40,42,44,45,43,41,39,37,35,33,31,29,27,25);(9,10,12,14,15,13,11)

Conjugacy classes & character table

8457 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33810, 80);