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C2×C112⋊C3 #10624393

order726
isoclinismisoclinic class of C112⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C11

Aliases

direct productC2×C112⋊C3
unambiguous semidirect product(C2×C112)⋊C3

Splittings

Fitting, socle, HallC2×C112 ⋊ C3
derived, SylowC112 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2C112⋊C3
commutator, cosocle, hypocenterC112C2×C3
FrattiniC1G
Fitting, socleC2×C112C3
LCSG ▷ C112
UCSC1 ◁ C2

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 66

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13089712, (C11⋊Dic11)⋊C3, C4×C112⋊C3, …
maximal normal subgroup of#13089712, (C11⋊Dic11)⋊C3, C4×C112⋊C3, …
maximal quotient ofC4×C112⋊C3, C22×C112⋊C3, C2×C112⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(726, 7);