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C3×#6396 #10620579

order672
isoclinismisoclinic class of #1141
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#6396

Splittings

HallC3×C7 ⋊ #108
SylowC7 ⋊ C3×#108

Subgroups

subgroupG/H
centerC3×C4C22⋊D7
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C2×C3×D7
FittingC3×C7×M16C2
socleC2×C3×C7C4⋊C4
prime radicalC22×C3×C7Q8
cosocleC2×C4×C7C22×C3
hypercenterC3×M16D7
hypocenterC7C3×#108
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×M16

60 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)(4,5,7,9,10,8,6)(11,13,14,22,16,23,15,24)(12,17,18,21,20,25,19,26);(5,6)(7,8)(9,10)(11,12,15,19,16,20,14,18)(13,21,22,25,23,26,24,17)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#6396, #13053549, #13053551, …
maximal normal subgroup ofD3×#6396, #13053549, #13053551, …
maximal quotient ofC3×#89274, C3×#89275, C3×#89353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 161);