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C2×C5×(C23⋊D23)⋊C3 #44540725

order31740
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2×C23

Aliases

direct productC2×C5×(C23⋊D23)⋊C3
unambiguous semidirect product(C2×C23⋊D23)⋊(C3×C5)
unambiguous semidirect product(C2×C5×C23⋊D23)⋊C3

Splittings

Fitting, socleC2×C5×C232 ⋊ C2×C3
derived, SylowC232 ⋊ C22×C3×C5
HallC2×C5×C23⋊D23 ⋊ C3
HallC5×C232⋊C3 ⋊ C22
HallC5×C232 ⋊ C22×C3
HallC2×C23⋊D23 ⋊ C3×C5
HallC232⋊C3 ⋊ C22×C5

Subgroups

subgroupG/H
center, hypercenterC2×C5(C23⋊D23)⋊C3
commutator, cosocle, hypocenterC232C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C232C2×C3
LCSG ▷ C232
UCSC1 ◁ C2×C5

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 76
transitive degreeμt(G) = 690

A minimal faithful permutation representation — 2 generators on 76 points:

(3,4,5,6,7)(8,9,11)(10,26,27,14,15,40)(12,37,39,19,20,57)(13,18,16,23,24,30)(17,56,21,32,33,28)(22,63,70,43,44,74)(25,36,55,45,48,62)(29,66,31,51,52,54)(34,71,49,47,58,42)(35,73,53,60,61,50)(38,41,75,46,64,67)(59,76,65,68,72,69);(1,2)(8,10,16,31,55,61,74,33,57,71,75,76,72,67,58,39,56,70,73,62,54,30,14)(9,12,21,26,49,43,23,46,53,51,65,25,45,69,29,50,38,13,22,42,15,28,19)(11,17,34,18,35,59,36,52,64,44,27,37,20,40,63,41,66,48,68,60,24,47,32)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31740, 63);