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C2×C3×C5×D561 #46778204

order33660
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×C5×D561
unambiguous semidirect product(C2×C3)⋊(C3×C5×D187)
unambiguous semidirect product(C2×C17)⋊(C3×C5×D33)

Splittings

Fitting, socleC2×C32×C5×C11×C17 ⋊ C2
derivedC3×C11×C17 ⋊ C22×C3×C5
HallC32×C5×C11×C17 ⋊ C22
HallC32×C11×C17 ⋊ C22×C5
HallC5×C11×C17 ⋊ C2×C3×D3
HallC32×C5×C17 ⋊ C2×D11
HallC32×C5×C11 ⋊ C2×D17
HallC11×C17 ⋊ C2×C3×C5×D3
HallC32×C17 ⋊ C2×C5×D11
HallC32×C11 ⋊ C2×C5×D17
HallC5×C17 ⋊ C2×C3×D33
HallC5×C11 ⋊ C2×C3×D51
HallC32×C5 ⋊ C2×D187
SylowC17 ⋊ C2×C3×C5×D33
SylowC11 ⋊ C2×C3×C5×D51
SylowC32 ⋊ C2×C5×D187

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5D561
commutator, cosocle, hypocenterC3×C11×C17C22×C3×C5
FrattiniC1G
Fitting, socleC2×C32×C5×C11×C17C2
LCSG ▷ C3×C11×C17
UCSC1 ◁ C2×C3×C5

176 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 41
transitive degreeμt(G) = 11220

A minimal faithful permutation representation — 2 generators on 41 points:

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

Conjugacy classes & character table

8460 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33660, 372);