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C5×C3⋊(C7⋊Dic77) #44568522

order32340
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C7

Aliases

direct productC5×C3⋊(C7⋊Dic77)
unambiguous semidirect productC3⋊(C5×C7⋊Dic77)
unambiguous semidirect productC7⋊(C5×Dic231)

Splittings

derived, HallC3×C72×C11 ⋊ C4×C5
HallC3×C5×C72×C11 ⋊ C4
HallC5×C72×C11 ⋊ Dic3
HallC3×C5×C72 ⋊ Dic11
HallC72×C11 ⋊ C5×Dic3
HallC5×C72 ⋊ Dic33
HallC3×C5×C11 ⋊ C7⋊Dic7
HallC3×C72 ⋊ C5×Dic11
HallC5×C11 ⋊ C3⋊(C7⋊Dic7)
SylowC72 ⋊ C5×Dic33
HallC3×C11 ⋊ C5×C7⋊Dic7
HallC3×C5 ⋊ C7⋊Dic77
SylowC11 ⋊ C5×C3⋊(C7⋊Dic7)
SylowC3 ⋊ C5×C7⋊Dic77

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊(C7⋊D77)
commutator, hypocenterC3×C72×C11C4×C5
FrattiniC2C5×C3⋊(C7⋊D77)
Fitting, socle, prime radicalC2×C3×C5×C72×C11C2
cosocleC2×C3×C72×C11C2×C5
LCSG ▷ C3×C72×C11
UCSC1 ◁ C2×C5

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 32340

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

8100 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 224);