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C3×#6393 #10620576

order672
isoclinismisoclinic class of #1132
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#6393

Splittings

HallC3×C7 ⋊ #108
SylowC7 ⋊ C3×#108

Subgroups

subgroupG/H
centerC3×C4D28
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C2×C3×D7
FittingC2×C3×C7×C8C2
socleC2×C3×C7C4⋊C4
prime radicalC22×C3×C7Q8
cosocleC2×C4×C7C22×C3
hypercenterC2×C3×C8D7
hypocenterC7C3×#108
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

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

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13050933, #13050935, #13050945, …
maximal normal subgroup of#13050933, #13050935, #13050945, …
maximal quotient ofC3×#89252, C3×#89253, C3×#89347, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 158);