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C3×D112 #10620556

order672
isoclinismisoclinic class of D112
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×D112

Splittings

FittingC3×C7×C16 ⋊ C2
HallC3×C7 ⋊ D16
SylowC7 ⋊ C3×D16

Subgroups

subgroupG/H
centerC2×C3D56
commutator, cosocleC7×C8C22×C3
FrattiniC8C2×C3×D7
FittingC3×C7×C16C2
socleC2×C3×C7D8
hypercenterC3×C16D7
hypocenterC7C3×D16
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×C16

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13050846, #13050847, #13050849, …
maximal normal subgroup of#13050846, #13050847, #13050849, …
maximal quotient ofC3×#89244, C3×D224, C3×Dic112, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 138);