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C2×C5⋊(C5⋊(#66526⋊C2)) #45686196

order33300
automorphism?
homologyAbelianization (H1) = C22×C9 · Schur multiplier (H2) = C2×C5

Aliases

direct productC2×C5⋊(C5⋊(#66526⋊C2))
unambiguous semidirect product(C2×C5)⋊(C5⋊(#66526⋊C2))
unambiguous semidirect product(C2×#66526)⋊(C5⋊D5)

Splittings

Fitting, socleC2×C52×C37 ⋊ C2×C9
derived, HallC52×C37 ⋊ C22×C9
HallC52×#66526 ⋊ C22
HallC2×C5⋊D185 ⋊ C9
Hall#66526 ⋊ C2×C5⋊D5
SylowC37 ⋊ C2×C9×C5⋊D5
SylowC52 ⋊ C2×#66526⋊C2

Subgroups

subgroupG/H
center, hypercenterC2C5⋊(C5⋊(#66526⋊C2))
commutator, hypocenterC52×C37C22×C9
FrattiniC1G
Fitting, socleC2×C52×C37C2×C9
prime radicalC2×C5⋊(C5⋊(D37⋊C3))C3
cosocleC52×C37⋊C3C22×C3
LCSG ▷ C52×C37
UCSC1 ◁ C2

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 1850

A minimal faithful permutation representation — 3 generators on 49 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 469);