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C2×C32×C5⋊D185 #45686513

order33300
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3×C5

Aliases

direct productC2×C32×C5⋊D185
unambiguous semidirect product(C2×C5)⋊(C32×D185)

Splittings

Fitting, socleC2×C32×C52×C37 ⋊ C2
derived, HallC52×C37 ⋊ C22×C32
HallC32×C52×C37 ⋊ C22
HallC32×C37 ⋊ C2×C5⋊D5
HallC32×C52 ⋊ C2×D37
SylowC37 ⋊ C2×C32×C5⋊D5
SylowC52 ⋊ C2×C32×D37

Subgroups

subgroupG/H
center, hypercenterC2×C32C5⋊D185
commutator, cosocle, hypocenterC52×C37C22×C32
FrattiniC1G
Fitting, socleC2×C32×C52×C37C2
LCSG ▷ C52×C37
UCSC1 ◁ C2×C32

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 16650

A minimal faithful permutation representation — 3 generators on 55 points:

(6,7,8)(10,11)(12,13)(15,16)(17,18)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55);(1,2)(3,4,5)(14,15,17,18,16)(19,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21);(9,10,12,13,11)

Conjugacy classes & character table

8352 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 786);