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C7×C3⋊D741 #44528041

order31122
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×C3⋊D741
unambiguous semidirect productC3⋊(C7×D741)
unambiguous semidirect productC13⋊(C7×C3⋊D57)
unambiguous semidirect productC19⋊(C7×C3⋊D39)

Splittings

Fitting, socle, HallC32×C7×C13×C19 ⋊ C2
derived, HallC32×C13×C19 ⋊ C2×C7
HallC7×C13×C19 ⋊ C3⋊D3
HallC32×C7×C19 ⋊ D13
HallC32×C7×C13 ⋊ D19
HallC13×C19 ⋊ C7×C3⋊D3
HallC32×C19 ⋊ C7×D13
HallC7×C19 ⋊ C3⋊D39
HallC32×C13 ⋊ C7×D19
HallC7×C13 ⋊ C3⋊D57
HallC32×C7 ⋊ D247
SylowC19 ⋊ C7×C3⋊D39
SylowC13 ⋊ C7×C3⋊D57
SylowC32 ⋊ C7×D247

Subgroups

subgroupG/H
center, hypercenterC7C3⋊D741
commutator, cosocle, hypocenterC32×C13×C19C2×C7
FrattiniC1G
Fitting, socleC32×C7×C13×C19C2
LCSG ▷ C32×C13×C19
UCSC1 ◁ C7

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 15561

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

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

Conjugacy classes & character table

7791 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31122, 524);