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G = C2×C31⋊C6order 372 = 22·3·31

Direct product of C2 and C31⋊C6

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C2×C31⋊C6, C62⋊C6, D62⋊C3, D31⋊C6, C31⋊(C2×C6), C31⋊C3⋊C22, (C2×C31⋊C3)⋊C2, SmallGroup(372,7)

Series: Derived Chief Lower central Upper central

C1C31 — C2×C31⋊C6
C1C31C31⋊C3C31⋊C6 — C2×C31⋊C6
C31 — C2×C31⋊C6
C1C2

Generators and relations for C2×C31⋊C6
 G = < a,b,c | a2=b31=c6=1, ab=ba, ac=ca, cbc-1=b6 >

31C2
31C2
31C3
31C22
31C6
31C6
31C6
31C2×C6

Character table of C2×C31⋊C6

 class 12A2B2C3A3B6A6B6C6D6E6F31A31B31C31D31E62A62B62C62D62E
 size 11313131313131313131316666666666
ρ11111111111111111111111    trivial
ρ21-11-111-111-1-1-111111-1-1-1-1-1    linear of order 2
ρ31-1-11111-1-1-11-111111-1-1-1-1-1    linear of order 2
ρ411-1-111-1-1-11-111111111111    linear of order 2
ρ51111ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ31111111111    linear of order 3
ρ611-1-1ζ3ζ32ζ65ζ6ζ65ζ3ζ6ζ321111111111    linear of order 6
ρ71-1-11ζ3ζ32ζ3ζ6ζ65ζ65ζ32ζ611111-1-1-1-1-1    linear of order 6
ρ81-1-11ζ32ζ3ζ32ζ65ζ6ζ6ζ3ζ6511111-1-1-1-1-1    linear of order 6
ρ91-11-1ζ3ζ32ζ65ζ32ζ3ζ65ζ6ζ611111-1-1-1-1-1    linear of order 6
ρ1011-1-1ζ32ζ3ζ6ζ65ζ6ζ32ζ65ζ31111111111    linear of order 6
ρ111-11-1ζ32ζ3ζ6ζ3ζ32ζ6ζ65ζ6511111-1-1-1-1-1    linear of order 6
ρ121111ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ321111111111    linear of order 3
ρ136-60000000000ζ3123312231173114319318ζ31293121311931123110312ζ3127312431203111317314ζ31283118311631153113313ζ3130312631253163153131293121311931123110312312731243120311131731431283118311631153113313313031263125316315313123312231173114319318    orthogonal faithful
ρ146-60000000000ζ3127312431203111317314ζ31303126312531631531ζ31293121311931123110312ζ3123312231173114319318ζ3128311831163115311331331303126312531631531312931213119311231103123123312231173114319318312831183116311531133133127312431203111317314    orthogonal faithful
ρ156-60000000000ζ31303126312531631531ζ3123312231173114319318ζ31283118311631153113313ζ31293121311931123110312ζ312731243120311131731431233122311731143193183128311831163115311331331293121311931123110312312731243120311131731431303126312531631531    orthogonal faithful
ρ16660000000000ζ31303126312531631531ζ3123312231173114319318ζ31283118311631153113313ζ31293121311931123110312ζ3127312431203111317314ζ3123312231173114319318ζ31283118311631153113313ζ31293121311931123110312ζ3127312431203111317314ζ31303126312531631531    orthogonal lifted from C31⋊C6
ρ17660000000000ζ31293121311931123110312ζ31283118311631153113313ζ31303126312531631531ζ3127312431203111317314ζ3123312231173114319318ζ31283118311631153113313ζ31303126312531631531ζ3127312431203111317314ζ3123312231173114319318ζ31293121311931123110312    orthogonal lifted from C31⋊C6
ρ18660000000000ζ3123312231173114319318ζ31293121311931123110312ζ3127312431203111317314ζ31283118311631153113313ζ31303126312531631531ζ31293121311931123110312ζ3127312431203111317314ζ31283118311631153113313ζ31303126312531631531ζ3123312231173114319318    orthogonal lifted from C31⋊C6
ρ19660000000000ζ31283118311631153113313ζ3127312431203111317314ζ3123312231173114319318ζ31303126312531631531ζ31293121311931123110312ζ3127312431203111317314ζ3123312231173114319318ζ31303126312531631531ζ31293121311931123110312ζ31283118311631153113313    orthogonal lifted from C31⋊C6
ρ206-60000000000ζ31293121311931123110312ζ31283118311631153113313ζ31303126312531631531ζ3127312431203111317314ζ312331223117311431931831283118311631153113313313031263125316315313127312431203111317314312331223117311431931831293121311931123110312    orthogonal faithful
ρ216-60000000000ζ31283118311631153113313ζ3127312431203111317314ζ3123312231173114319318ζ31303126312531631531ζ3129312131193112311031231273124312031113173143123312231173114319318313031263125316315313129312131193112311031231283118311631153113313    orthogonal faithful
ρ22660000000000ζ3127312431203111317314ζ31303126312531631531ζ31293121311931123110312ζ3123312231173114319318ζ31283118311631153113313ζ31303126312531631531ζ31293121311931123110312ζ3123312231173114319318ζ31283118311631153113313ζ3127312431203111317314    orthogonal lifted from C31⋊C6

Smallest permutation representation of C2×C31⋊C6
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 27 26 31 6 7)(3 22 20 30 11 13)(4 17 14 29 16 19)(5 12 8 28 21 25)(9 23 15 24 10 18)(33 58 57 62 37 38)(34 53 51 61 42 44)(35 48 45 60 47 50)(36 43 39 59 52 56)(40 54 46 55 41 49)

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

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

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

Matrix representation of C2×C31⋊C6 in GL6(𝔽373)

37200000
03720000
00372000
00037200
00003720
00000372
,
323363144363323372
100000
010000
001000
000100
000010
,
100000
372323363144363323
10050204366271109
1601716916917160
10927136620450100
323363144363323372

G:=sub<GL(6,GF(373))| [372,0,0,0,0,0,0,372,0,0,0,0,0,0,372,0,0,0,0,0,0,372,0,0,0,0,0,0,372,0,0,0,0,0,0,372],[323,1,0,0,0,0,363,0,1,0,0,0,144,0,0,1,0,0,363,0,0,0,1,0,323,0,0,0,0,1,372,0,0,0,0,0],[1,372,100,160,109,323,0,323,50,17,271,363,0,363,204,169,366,144,0,144,366,169,204,363,0,363,271,17,50,323,0,323,109,160,100,372] >;

C2×C31⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_{31}\rtimes C_6
% in TeX

G:=Group("C2xC31:C6");
// GroupNames label

G:=SmallGroup(372,7);
// by ID

G=gap.SmallGroup(372,7);
# by ID

G:=PCGroup([4,-2,-2,-3,-31,5763,1211]);
// Polycyclic

G:=Group<a,b,c|a^2=b^31=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations

Export

Subgroup lattice of C2×C31⋊C6 in TeX
Character table of C2×C31⋊C6 in TeX

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