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C3×D41×(C5⋊D5)⋊C3 #49146995

order36900
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3×C5

Aliases

direct productC3×D41×(C5⋊D5)⋊C3
unambiguous semidirect product(C5⋊D5×D41)⋊C32
unambiguous semidirect product(C3×C5⋊D5×D41)⋊C3

Splittings

Fitting, socleC3×C52×C41 ⋊ C22×C3
derived, HallC52×C41 ⋊ C22×C32
HallC3×C41×C52⋊C3 ⋊ C22
HallC5⋊D5×D41 ⋊ C32
HallC3×C52⋊C3 ⋊ C2×D41
SylowC41 ⋊ C2×C3×(C5⋊D5)⋊C3
SylowC52 ⋊ C2×C32×D41

Subgroups

subgroupG/H
center, hypercenterC3D41×(C5⋊D5)⋊C3
commutator, cosocle, hypocenterC52×C41C22×C32
FrattiniC1G
Fitting, socleC3×C52×C41C22×C3
LCSG ▷ C52×C41
UCSC1 ◁ C3

160 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 59
transitive degreeμt(G) = 1845

A minimal faithful permutation representation — 2 generators on 59 points:

(4,5,7)(6,9,13,10,8,15)(11,16,18,17,14,12)(19,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21);(1,2,3)(4,6,11,17,10)(5,8,14,16,9)(7,12,13,15,18)(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)(56,57)(58,59)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36900, 396);