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C2×C3⋊D1113 #27211326

order13356
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3⋊D1113
unambiguous semidirect product(C2×C3)⋊D1113
unambiguous semidirect product(C2×C7)⋊(C3⋊D159)
unambiguous semidirect product(C2×C53)⋊(C3⋊D21)

Splittings

Fitting, socleC2×C32×C7×C53 ⋊ C2
derived, HallC32×C7×C53 ⋊ C22
HallC32×C53 ⋊ C2×D7
HallC7×C53 ⋊ C2×C3⋊D3
HallC32×C7 ⋊ C2×D53
SylowC53 ⋊ C2×C3⋊D21
SylowC32 ⋊ C2×D371
SylowC7 ⋊ C2×C3⋊D159

Subgroups

subgroupG/H
center, hypercenterC2C3⋊D1113
commutator, cosocle, hypocenterC32×C7×C53C22
FrattiniC1G
Fitting, socleC2×C32×C7×C53C2
LCSG ▷ C32×C7×C53
UCSC1 ◁ C2

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 6678

A minimal faithful permutation representation — 3 generators on 68 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(55,56)(57,58)(59,60)(62,63)(65,66);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,57,59,60,58,56)(64,65,66)(67,68);(61,62,63)

Conjugacy classes & character table

3342 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 119);