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C3×#23778464 #44477937

order30420
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×#23778464

Splittings

Fitting, socleC3×C5×C132 ⋊ C3×C4
derived, HallC5×C132 ⋊ C32×C4
HallC3×C5×C13×C13⋊C3 ⋊ C4
Hall#16810003 ⋊ C32
HallC3×C13×C13⋊C3 ⋊ AGL(1,𝔽5)
SylowC132 ⋊ C32×AGL(1,𝔽5)
SylowC5 ⋊ C3×#14523150

Subgroups

subgroupG/H
center, hypercenterC3#23778464
commutator, hypocenterC5×C132C32×C4
FrattiniC1G
Fitting, socleC3×C5×C132C3×C4
prime radicalC3×C5⋊((C13⋊C3)⋊D13)C2
cosocleC5⋊(C13⋊D13)C2×C32
LCSG ▷ C5×C132
UCSC1 ◁ C3

192 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 34
transitive degreeμt(G) = 585

A minimal faithful permutation representation — 2 generators on 34 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 526);