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C2×C7×C3⋊D345 #44394682

order28980
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C7×C3⋊D345
unambiguous semidirect product(C2×C3)⋊(C7×D345)
unambiguous semidirect product(C2×C5)⋊(C7×C3⋊D69)
unambiguous semidirect product(C2×C23)⋊(C7×C3⋊D15)

Splittings

Fitting, socleC2×C32×C5×C7×C23 ⋊ C2
derived, HallC32×C5×C23 ⋊ C22×C7
HallC32×C5×C7×C23 ⋊ C22
HallC32×C7×C23 ⋊ C2×D5
HallC5×C7×C23 ⋊ C2×C3⋊D3
HallC32×C5×C7 ⋊ C2×D23
HallC32×C23 ⋊ C2×C7×D5
HallC7×C23 ⋊ C2×C3⋊D15
HallC5×C23 ⋊ C2×C7×C3⋊D3
HallC32×C7 ⋊ C2×D115
HallC32×C5 ⋊ C2×C7×D23
HallC5×C7 ⋊ C2×C3⋊D69
SylowC23 ⋊ C2×C7×C3⋊D15
SylowC32 ⋊ C2×C7×D115
SylowC5 ⋊ C2×C7×C3⋊D69

Subgroups

subgroupG/H
center, hypercenterC2×C7C3⋊D345
commutator, cosocle, hypocenterC32×C5×C23C22×C7
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C23C2
LCSG ▷ C32×C5×C23
UCSC1 ◁ C2×C7

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 14490

A minimal faithful permutation representation — 3 generators on 43 points:

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

Conjugacy classes & character table

7266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28980, 351);