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G = C31⋊C12order 372 = 22·3·31

The semidirect product of C31 and C12 acting via C12/C2=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C31⋊C12, C62.C6, Dic31⋊C3, C31⋊C3⋊C4, C2.(C31⋊C6), (C2×C31⋊C3).C2, SmallGroup(372,1)

Series: Derived Chief Lower central Upper central

C1C31 — C31⋊C12
C1C31C62C2×C31⋊C3 — C31⋊C12
C31 — C31⋊C12
C1C2

Generators and relations for C31⋊C12
 G = < a,b | a31=b12=1, bab-1=a26 >

31C3
31C4
31C6
31C12

Character table of C31⋊C12

 class 123A3B4A4B6A6B12A12B12C12D31A31B31C31D31E62A62B62C62D62E
 size 11313131313131313131316666666666
ρ11111111111111111111111    trivial
ρ21111-1-111-1-1-1-11111111111    linear of order 2
ρ311ζ3ζ32-1-1ζ32ζ3ζ65ζ6ζ65ζ61111111111    linear of order 6
ρ411ζ3ζ3211ζ32ζ3ζ3ζ32ζ3ζ321111111111    linear of order 3
ρ511ζ32ζ3-1-1ζ3ζ32ζ6ζ65ζ6ζ651111111111    linear of order 6
ρ611ζ32ζ311ζ3ζ32ζ32ζ3ζ32ζ31111111111    linear of order 3
ρ71-111-ii-1-1ii-i-i11111-1-1-1-1-1    linear of order 4
ρ81-111i-i-1-1-i-iii11111-1-1-1-1-1    linear of order 4
ρ91-1ζ32ζ3i-iζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ311111-1-1-1-1-1    linear of order 12
ρ101-1ζ3ζ32-iiζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ3211111-1-1-1-1-1    linear of order 12
ρ111-1ζ32ζ3-iiζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ311111-1-1-1-1-1    linear of order 12
ρ121-1ζ3ζ32i-iζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ3211111-1-1-1-1-1    linear of order 12
ρ13660000000000ζ3127312431203111317314ζ31303126312531631531ζ3123312231173114319318ζ31283118311631153113313ζ31293121311931123110312ζ31283118311631153113313ζ3127312431203111317314ζ31303126312531631531ζ3123312231173114319318ζ31293121311931123110312    orthogonal lifted from C31⋊C6
ρ14660000000000ζ3123312231173114319318ζ31293121311931123110312ζ31283118311631153113313ζ31303126312531631531ζ3127312431203111317314ζ31303126312531631531ζ3123312231173114319318ζ31293121311931123110312ζ31283118311631153113313ζ3127312431203111317314    orthogonal lifted from C31⋊C6
ρ15660000000000ζ31293121311931123110312ζ31283118311631153113313ζ3127312431203111317314ζ3123312231173114319318ζ31303126312531631531ζ3123312231173114319318ζ31293121311931123110312ζ31283118311631153113313ζ3127312431203111317314ζ31303126312531631531    orthogonal lifted from C31⋊C6
ρ16660000000000ζ31283118311631153113313ζ3127312431203111317314ζ31303126312531631531ζ31293121311931123110312ζ3123312231173114319318ζ31293121311931123110312ζ31283118311631153113313ζ3127312431203111317314ζ31303126312531631531ζ3123312231173114319318    orthogonal lifted from C31⋊C6
ρ17660000000000ζ31303126312531631531ζ3123312231173114319318ζ31293121311931123110312ζ3127312431203111317314ζ31283118311631153113313ζ3127312431203111317314ζ31303126312531631531ζ3123312231173114319318ζ31293121311931123110312ζ31283118311631153113313    orthogonal lifted from C31⋊C6
ρ186-60000000000ζ3123312231173114319318ζ31293121311931123110312ζ31283118311631153113313ζ31303126312531631531ζ312731243120311131731431303126312531631531312331223117311431931831293121311931123110312312831183116311531133133127312431203111317314    symplectic faithful, Schur index 2
ρ196-60000000000ζ3127312431203111317314ζ31303126312531631531ζ3123312231173114319318ζ31283118311631153113313ζ3129312131193112311031231283118311631153113313312731243120311131731431303126312531631531312331223117311431931831293121311931123110312    symplectic faithful, Schur index 2
ρ206-60000000000ζ31293121311931123110312ζ31283118311631153113313ζ3127312431203111317314ζ3123312231173114319318ζ3130312631253163153131233122311731143193183129312131193112311031231283118311631153113313312731243120311131731431303126312531631531    symplectic faithful, Schur index 2
ρ216-60000000000ζ31283118311631153113313ζ3127312431203111317314ζ31303126312531631531ζ31293121311931123110312ζ312331223117311431931831293121311931123110312312831183116311531133133127312431203111317314313031263125316315313123312231173114319318    symplectic faithful, Schur index 2
ρ226-60000000000ζ31303126312531631531ζ3123312231173114319318ζ31293121311931123110312ζ3127312431203111317314ζ3128311831163115311331331273124312031113173143130312631253163153131233122311731143193183129312131193112311031231283118311631153113313    symplectic faithful, Schur index 2

Smallest permutation representation of C31⋊C12
On 124 points
Generators in S124
(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)(63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93)(94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124)
(1 94 35 63)(2 100 40 93 26 120 36 69 6 124 60 89)(3 106 45 92 20 115 37 75 11 123 54 84)(4 112 50 91 14 110 38 81 16 122 48 79)(5 118 55 90 8 105 39 87 21 121 42 74)(7 99 34 88 27 95 41 68 31 119 61 64)(9 111 44 86 15 116 43 80 10 117 49 85)(12 98 59 83 28 101 46 67 25 114 62 70)(13 104 33 82 22 96 47 73 30 113 56 65)(17 97 53 78 29 107 51 66 19 109 32 76)(18 103 58 77 23 102 52 72 24 108 57 71)

G:=sub<Sym(124)| (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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,94,35,63)(2,100,40,93,26,120,36,69,6,124,60,89)(3,106,45,92,20,115,37,75,11,123,54,84)(4,112,50,91,14,110,38,81,16,122,48,79)(5,118,55,90,8,105,39,87,21,121,42,74)(7,99,34,88,27,95,41,68,31,119,61,64)(9,111,44,86,15,116,43,80,10,117,49,85)(12,98,59,83,28,101,46,67,25,114,62,70)(13,104,33,82,22,96,47,73,30,113,56,65)(17,97,53,78,29,107,51,66,19,109,32,76)(18,103,58,77,23,102,52,72,24,108,57,71)>;

G:=Group( (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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,94,35,63)(2,100,40,93,26,120,36,69,6,124,60,89)(3,106,45,92,20,115,37,75,11,123,54,84)(4,112,50,91,14,110,38,81,16,122,48,79)(5,118,55,90,8,105,39,87,21,121,42,74)(7,99,34,88,27,95,41,68,31,119,61,64)(9,111,44,86,15,116,43,80,10,117,49,85)(12,98,59,83,28,101,46,67,25,114,62,70)(13,104,33,82,22,96,47,73,30,113,56,65)(17,97,53,78,29,107,51,66,19,109,32,76)(18,103,58,77,23,102,52,72,24,108,57,71) );

G=PermutationGroup([(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),(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93),(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(1,94,35,63),(2,100,40,93,26,120,36,69,6,124,60,89),(3,106,45,92,20,115,37,75,11,123,54,84),(4,112,50,91,14,110,38,81,16,122,48,79),(5,118,55,90,8,105,39,87,21,121,42,74),(7,99,34,88,27,95,41,68,31,119,61,64),(9,111,44,86,15,116,43,80,10,117,49,85),(12,98,59,83,28,101,46,67,25,114,62,70),(13,104,33,82,22,96,47,73,30,113,56,65),(17,97,53,78,29,107,51,66,19,109,32,76),(18,103,58,77,23,102,52,72,24,108,57,71)])

Matrix representation of C31⋊C12 in GL7(𝔽373)

1000000
037210000
037201000
037200100
037200010
037200001
03141397429923458
,
69000000
0105272125264342
068231371276323150
086116685563358
0249255347103366273
0315295293295315279
022637136266368297

G:=sub<GL(7,GF(373))| [1,0,0,0,0,0,0,0,372,372,372,372,372,314,0,1,0,0,0,0,139,0,0,1,0,0,0,74,0,0,0,1,0,0,299,0,0,0,0,1,0,234,0,0,0,0,0,1,58],[69,0,0,0,0,0,0,0,105,68,86,249,315,226,0,272,231,116,255,295,37,0,125,371,68,347,293,136,0,26,276,55,103,295,266,0,43,323,63,366,315,368,0,42,150,358,273,279,297] >;

C31⋊C12 in GAP, Magma, Sage, TeX

C_{31}\rtimes C_{12}
% in TeX

G:=Group("C31:C12");
// GroupNames label

G:=SmallGroup(372,1);
// by ID

G=gap.SmallGroup(372,1);
# by ID

G:=PCGroup([4,-2,-3,-2,-31,24,5763,2407]);
// Polycyclic

G:=Group<a,b|a^31=b^12=1,b*a*b^-1=a^26>;
// generators/relations

Export

Subgroup lattice of C31⋊C12 in TeX
Character table of C31⋊C12 in TeX

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