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C5×C7⋊(C112⋊C3) #27163201

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

Aliases

direct productC5×C7⋊(C112⋊C3)
unambiguous semidirect productC7⋊(C5×C112⋊C3)
unambiguous semidirect product(C5×C7)⋊(C112⋊C3)
unambiguous semidirect product(C5×C112)⋊ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC5×C7×C112 ⋊ C3
derived, HallC7×C112 ⋊ C3×C5
HallC5×C112 ⋊ ΓL(1,𝔽8)
SylowC112 ⋊ C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C112⋊C3
SylowC7 ⋊ C5×C112⋊C3

Subgroups

subgroupG/H
center, hypercenterC5C7⋊(C112⋊C3)
commutator, cosocle, hypocenterC7×C112C3×C5
FrattiniC1G
Fitting, socleC5×C7×C112C3
LCSG ▷ C7×C112
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 1155

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

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

Conjugacy classes & character table

1425 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×D7⋊(C112⋊C3), C5×#19099620, C5×(C11⋊D11)⋊ΓL(1,𝔽8), …
maximal normal subgroup ofC5×D7⋊(C112⋊C3), C5×#19099620, C5×(C11⋊D11)⋊ΓL(1,𝔽8), …
maximal quotient ofC2×C5×C7⋊(C112⋊C3), C5×C7⋊(C112⋊C9), C3×C5×C7⋊(C112⋊C3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12705, 16);