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C3×#6482 #10621230

order672
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#6482

Splittings

FittingC3×C7×Q16 ⋊ C2
HallC3×C7 ⋊ #135
SylowC7 ⋊ C3×#135

Subgroups

subgroupG/H
centerC2×C3D4×D7
commutator, cosocleC4×C7C23×C3
FrattiniC4C22×C3×D7
FittingC3×C7×Q16C2
socleC2×C3×C7C2×D4
hypercenterC3×Q16D7
hypocenterC7C3×#135
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q16

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13053668, #13053674, #13053680, …
maximal normal subgroup of#13053668, #13053674, #13053680, …
maximal quotient ofC3×#89591, C3×#89568, C3×#89572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 812);