Finite Group Databasebeta

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C2×C3×C7×C3⋊(C11⋊D11) #44479264

order30492
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2×C11

Aliases

direct productC2×C3×C7×C3⋊(C11⋊D11)
unambiguous semidirect product(C2×C3)⋊(C3×C7×C11⋊D11)
unambiguous semidirect product(C2×C11)⋊(C3×C7×D33)

Splittings

Fitting, socleC2×C32×C7×C112 ⋊ C2
derivedC3×C112 ⋊ C22×C3×C7
HallC32×C7×C112 ⋊ C22
HallC32×C112 ⋊ C22×C7
HallC7×C112 ⋊ C2×C3×D3
SylowC112 ⋊ C2×C3×D3×C7
HallC32×C7 ⋊ C2×C11⋊D11
SylowC32 ⋊ C2×C7×C11⋊D11

Subgroups

subgroupG/H
center, hypercenterC2×C3×C7C3⋊(C11⋊D11)
commutator, cosocle, hypocenterC3×C112C22×C3×C7
FrattiniC1G
Fitting, socleC2×C32×C7×C112C2
LCSG ▷ C3×C112
UCSC1 ◁ C2×C3×C7

616 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 236 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) = 37
transitive degreeμt(G) = 10164

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

7686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 487);