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G = C2×C31⋊C3order 186 = 2·3·31

Direct product of C2 and C31⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C2×C31⋊C3, C62⋊C3, C312C6, SmallGroup(186,2)

Series: Derived Chief Lower central Upper central

C1C31 — C2×C31⋊C3
C1C31C31⋊C3 — C2×C31⋊C3
C31 — C2×C31⋊C3
C1C2

Generators and relations for C2×C31⋊C3
 G = < a,b,c | a2=b31=c3=1, ab=ba, ac=ca, cbc-1=b5 >

31C3
31C6

Character table of C2×C31⋊C3

 class 123A3B6A6B31A31B31C31D31E31F31G31H31I31J62A62B62C62D62E62F62G62H62I62J
 size 113131313133333333333333333333
ρ111111111111111111111111111    trivial
ρ21-111-1-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ3211111111111111111111    linear of order 3
ρ41-1ζ32ζ3ζ65ζ61111111111-1-1-1-1-1-1-1-1-1-1    linear of order 6
ρ51-1ζ3ζ32ζ6ζ651111111111-1-1-1-1-1-1-1-1-1-1    linear of order 6
ρ611ζ3ζ32ζ32ζ311111111111111111111    linear of order 3
ρ7330000ζ312331223117ζ312831183116ζ3114319318ζ312931213112ζ312531531ζ31193110312ζ31153113313ζ312731243111ζ3120317314ζ31303126316ζ31193110312ζ31153113313ζ3120317314ζ31303126316ζ312831183116ζ3114319318ζ312931213112ζ312531531ζ312731243111ζ312331223117    complex lifted from C31⋊C3
ρ8330000ζ3120317314ζ312331223117ζ312731243111ζ312531531ζ31153113313ζ31303126316ζ3114319318ζ31193110312ζ312931213112ζ312831183116ζ31303126316ζ3114319318ζ312931213112ζ312831183116ζ312331223117ζ312731243111ζ312531531ζ31153113313ζ31193110312ζ3120317314    complex lifted from C31⋊C3
ρ9330000ζ312931213112ζ3120317314ζ31193110312ζ31153113313ζ3114319318ζ312831183116ζ312731243111ζ31303126316ζ312531531ζ312331223117ζ312831183116ζ312731243111ζ312531531ζ312331223117ζ3120317314ζ31193110312ζ31153113313ζ3114319318ζ31303126316ζ312931213112    complex lifted from C31⋊C3
ρ10330000ζ312831183116ζ31303126316ζ31153113313ζ3120317314ζ312931213112ζ312731243111ζ312531531ζ3114319318ζ312331223117ζ31193110312ζ312731243111ζ312531531ζ312331223117ζ31193110312ζ31303126316ζ31153113313ζ3120317314ζ312931213112ζ3114319318ζ312831183116    complex lifted from C31⋊C3
ρ113-30000ζ3120317314ζ312331223117ζ312731243111ζ312531531ζ31153113313ζ31303126316ζ3114319318ζ31193110312ζ312931213112ζ31283118311631303126316311431931831293121311231283118311631233122311731273124311131253153131153113313311931103123120317314    complex faithful
ρ12330000ζ31193110312ζ312731243111ζ312931213112ζ312831183116ζ312331223117ζ31153113313ζ3120317314ζ312531531ζ31303126316ζ3114319318ζ31153113313ζ3120317314ζ31303126316ζ3114319318ζ312731243111ζ312931213112ζ312831183116ζ312331223117ζ312531531ζ31193110312    complex lifted from C31⋊C3
ρ133-30000ζ312531531ζ312931213112ζ31303126316ζ3114319318ζ312731243111ζ312331223117ζ31193110312ζ312831183116ζ31153113313ζ312031731431233122311731193110312311531133133120317314312931213112313031263163114319318312731243111312831183116312531531    complex faithful
ρ14330000ζ312731243111ζ3114319318ζ3120317314ζ31303126316ζ312831183116ζ312531531ζ312331223117ζ312931213112ζ31193110312ζ31153113313ζ312531531ζ312331223117ζ31193110312ζ31153113313ζ3114319318ζ3120317314ζ31303126316ζ312831183116ζ312931213112ζ312731243111    complex lifted from C31⋊C3
ρ153-30000ζ312931213112ζ3120317314ζ31193110312ζ31153113313ζ3114319318ζ312831183116ζ312731243111ζ31303126316ζ312531531ζ31233122311731283118311631273124311131253153131233122311731203173143119311031231153113313311431931831303126316312931213112    complex faithful
ρ16330000ζ3114319318ζ31153113313ζ312331223117ζ31193110312ζ31303126316ζ312931213112ζ312831183116ζ3120317314ζ312731243111ζ312531531ζ312931213112ζ312831183116ζ312731243111ζ312531531ζ31153113313ζ312331223117ζ31193110312ζ31303126316ζ3120317314ζ3114319318    complex lifted from C31⋊C3
ρ17330000ζ31303126316ζ31193110312ζ312531531ζ312331223117ζ3120317314ζ3114319318ζ312931213112ζ31153113313ζ312831183116ζ312731243111ζ3114319318ζ312931213112ζ312831183116ζ312731243111ζ31193110312ζ312531531ζ312331223117ζ3120317314ζ31153113313ζ31303126316    complex lifted from C31⋊C3
ρ183-30000ζ312331223117ζ312831183116ζ3114319318ζ312931213112ζ312531531ζ31193110312ζ31153113313ζ312731243111ζ3120317314ζ3130312631631193110312311531133133120317314313031263163128311831163114319318312931213112312531531312731243111312331223117    complex faithful
ρ193-30000ζ31193110312ζ312731243111ζ312931213112ζ312831183116ζ312331223117ζ31153113313ζ3120317314ζ312531531ζ31303126316ζ311431931831153113313312031731431303126316311431931831273124311131293121311231283118311631233122311731253153131193110312    complex faithful
ρ203-30000ζ31153113313ζ312531531ζ312831183116ζ312731243111ζ31193110312ζ3120317314ζ31303126316ζ312331223117ζ3114319318ζ31293121311231203173143130312631631143193183129312131123125315313128311831163127312431113119311031231233122311731153113313    complex faithful
ρ213-30000ζ31303126316ζ31193110312ζ312531531ζ312331223117ζ3120317314ζ3114319318ζ312931213112ζ31153113313ζ312831183116ζ31273124311131143193183129312131123128311831163127312431113119311031231253153131233122311731203173143115311331331303126316    complex faithful
ρ223-30000ζ312731243111ζ3114319318ζ3120317314ζ31303126316ζ312831183116ζ312531531ζ312331223117ζ312931213112ζ31193110312ζ3115311331331253153131233122311731193110312311531133133114319318312031731431303126316312831183116312931213112312731243111    complex faithful
ρ233-30000ζ312831183116ζ31303126316ζ31153113313ζ3120317314ζ312931213112ζ312731243111ζ312531531ζ3114319318ζ312331223117ζ3119311031231273124311131253153131233122311731193110312313031263163115311331331203173143129312131123114319318312831183116    complex faithful
ρ24330000ζ31153113313ζ312531531ζ312831183116ζ312731243111ζ31193110312ζ3120317314ζ31303126316ζ312331223117ζ3114319318ζ312931213112ζ3120317314ζ31303126316ζ3114319318ζ312931213112ζ312531531ζ312831183116ζ312731243111ζ31193110312ζ312331223117ζ31153113313    complex lifted from C31⋊C3
ρ25330000ζ312531531ζ312931213112ζ31303126316ζ3114319318ζ312731243111ζ312331223117ζ31193110312ζ312831183116ζ31153113313ζ3120317314ζ312331223117ζ31193110312ζ31153113313ζ3120317314ζ312931213112ζ31303126316ζ3114319318ζ312731243111ζ312831183116ζ312531531    complex lifted from C31⋊C3
ρ263-30000ζ3114319318ζ31153113313ζ312331223117ζ31193110312ζ31303126316ζ312931213112ζ312831183116ζ3120317314ζ312731243111ζ31253153131293121311231283118311631273124311131253153131153113313312331223117311931103123130312631631203173143114319318    complex faithful

Smallest permutation representation of C2×C31⋊C3
On 62 points
Generators in S62
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)
(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 27 28 29 30 31)(32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62)
(2 26 6)(3 20 11)(4 14 16)(5 8 21)(7 27 31)(9 15 10)(12 28 25)(13 22 30)(17 29 19)(18 23 24)(33 57 37)(34 51 42)(35 45 47)(36 39 52)(38 58 62)(40 46 41)(43 59 56)(44 53 61)(48 60 50)(49 54 55)

G:=sub<Sym(62)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)>;

G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55) );

G=PermutationGroup([(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31),(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62)], [(2,26,6),(3,20,11),(4,14,16),(5,8,21),(7,27,31),(9,15,10),(12,28,25),(13,22,30),(17,29,19),(18,23,24),(33,57,37),(34,51,42),(35,45,47),(36,39,52),(38,58,62),(40,46,41),(43,59,56),(44,53,61),(48,60,50),(49,54,55)])

C2×C31⋊C3 is a maximal subgroup of   C31⋊C12

Matrix representation of C2×C31⋊C3 in GL3(𝔽5) generated by

400
040
004
,
132
124
012
,
141
001
044
G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[1,1,0,3,2,1,2,4,2],[1,0,0,4,0,4,1,1,4] >;

C2×C31⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{31}\rtimes C_3
% in TeX

G:=Group("C2xC31:C3");
// GroupNames label

G:=SmallGroup(186,2);
// by ID

G=gap.SmallGroup(186,2);
# by ID

G:=PCGroup([3,-2,-3,-31,680]);
// Polycyclic

G:=Group<a,b,c|a^2=b^31=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C2×C31⋊C3 in TeX
Character table of C2×C31⋊C3 in TeX

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