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C2×#30659271 #45686189

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

Aliases

direct productC2×#30659271

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2#30659271
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

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 1110

A minimal faithful permutation representation — 2 generators on 54 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 462);