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G = C31⋊C6order 186 = 2·3·31

The semidirect product of C31 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C31⋊C6, D31⋊C3, C31⋊C3⋊C2, SmallGroup(186,1)

Series: Derived Chief Lower central Upper central

C1C31 — C31⋊C6
C1C31C31⋊C3 — C31⋊C6
C31 — C31⋊C6
C1

Generators and relations for C31⋊C6
 G = < a,b | a31=b6=1, bab-1=a6 >

31C2
31C3
31C6

Character table of C31⋊C6

 class 123A3B6A6B31A31B31C31D31E
 size 1313131313166666
ρ111111111111    trivial
ρ21-111-1-111111    linear of order 2
ρ31-1ζ3ζ32ζ65ζ611111    linear of order 6
ρ411ζ3ζ32ζ3ζ3211111    linear of order 3
ρ51-1ζ32ζ3ζ6ζ6511111    linear of order 6
ρ611ζ32ζ3ζ32ζ311111    linear of order 3
ρ7600000ζ31303126312531631531ζ31293121311931123110312ζ31283118311631153113313ζ3127312431203111317314ζ3123312231173114319318    orthogonal faithful
ρ8600000ζ3123312231173114319318ζ31283118311631153113313ζ3127312431203111317314ζ31303126312531631531ζ31293121311931123110312    orthogonal faithful
ρ9600000ζ31293121311931123110312ζ3127312431203111317314ζ31303126312531631531ζ3123312231173114319318ζ31283118311631153113313    orthogonal faithful
ρ10600000ζ31283118311631153113313ζ31303126312531631531ζ3123312231173114319318ζ31293121311931123110312ζ3127312431203111317314    orthogonal faithful
ρ11600000ζ3127312431203111317314ζ3123312231173114319318ζ31293121311931123110312ζ31283118311631153113313ζ31303126312531631531    orthogonal faithful

Permutation representations of C31⋊C6
On 31 points: primitive - transitive group 31T5
Generators in S31
(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)
(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)

G:=sub<Sym(31)| (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), (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)>;

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), (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) );

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)], [(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)])

G:=TransitiveGroup(31,5);

C31⋊C6 is a maximal quotient of   C31⋊C12

Matrix representation of C31⋊C6 in GL6(𝔽373)

010000
001000
000100
000010
000001
3725729121729157
,
100000
3725729121729157
1952527278328276
22043653654220
2763287872252195
5729121729157372

G:=sub<GL(6,GF(373))| [0,0,0,0,0,372,1,0,0,0,0,57,0,1,0,0,0,291,0,0,1,0,0,217,0,0,0,1,0,291,0,0,0,0,1,57],[1,372,195,220,276,57,0,57,252,4,328,291,0,291,72,365,78,217,0,217,78,365,72,291,0,291,328,4,252,57,0,57,276,220,195,372] >;

C31⋊C6 in GAP, Magma, Sage, TeX

C_{31}\rtimes C_6
% in TeX

G:=Group("C31:C6");
// GroupNames label

G:=SmallGroup(186,1);
// by ID

G=gap.SmallGroup(186,1);
# by ID

G:=PCGroup([3,-2,-3,-31,1622,680]);
// Polycyclic

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

Export

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

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