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C2×C3×#19099620 #44479145

order30492
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3×C11

Aliases

direct productC2×C3×#19099620

Splittings

Fitting, socleC2×C3×C7×C112 ⋊ C2×C3
derived, HallC7×C112 ⋊ C22×C32
HallC3×C7⋊(C112⋊C3) ⋊ C22
HallC2×C7⋊(C11⋊D11) ⋊ C32
SylowC112 ⋊ C2×C3×AGL(1,𝔽7)
SylowC7 ⋊ C2×C3×(C11⋊D11)⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C3#19099620
commutator, cosocle, hypocenterC7×C112C22×C32
FrattiniC1G
Fitting, socleC2×C3×C7×C112C2×C3
LCSG ▷ C7×C112
UCSC1 ◁ C2×C3

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 1386

A minimal faithful permutation representation — 2 generators on 45 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 368);