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C3×C7×#10597245 #27163200

order12705
automorphism?
homologyAbelianization (H1) = C3×C5×C7 · Schur multiplier (H2) = C11

Aliases

direct productC3×C7×#10597245

Splittings

Fitting, socle, HallC3×C7×C112 ⋊ C5
derived, SylowC112 ⋊ C3×C5×C7
HallC7×C112 ⋊ C3×C5
HallC3×C112 ⋊ C5×C7

Subgroups

subgroupG/H
center, hypercenterC3×C7#10597245
commutator, cosocle, hypocenterC112C3×C5×C7
FrattiniC1G
Fitting, socleC3×C7×C112C5
LCSG ▷ C112
UCSC1 ◁ C3×C7

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 1155

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

derived subgroup ofC3⋊(C7⋊#11925890)
maximal subgroup ofC3×C7×#11925890, C3×C7×#11925896, C3×C7×#11925893, …
maximal normal subgroup ofC3×C7×#11925890, C3×C7×#11925896, C3×C7×#11925893, …
maximal quotient ofC2×C3×C7×#10597245, C7×C9×#10597245, C32×C7×#10597245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12705, 15);