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C2×C3×#19099626 #44479154

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

Aliases

direct productC2×C3×#19099626

Splittings

Fitting, socleC2×C3×C7×C112 ⋊ C2×C3
derived, HallC7×C112 ⋊ C22×C32
HallC3×C7×C112⋊C3 ⋊ C22
HallC2×C7⋊(C11⋊D11) ⋊ C32
HallC3×C112⋊C3 ⋊ C2×D7
SylowC112 ⋊ C2×C32×D7
SylowC7 ⋊ C2×C3×(C11⋊D11)⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C3#19099626
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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 377);