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G = C61⋊C4order 244 = 22·61

The semidirect product of C61 and C4 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C61⋊C4, D61.C2, SmallGroup(244,3)

Series: Derived Chief Lower central Upper central

C1C61 — C61⋊C4
C1C61D61 — C61⋊C4
C61 — C61⋊C4
C1

Generators and relations for C61⋊C4
 G = < a,b | a61=b4=1, bab-1=a50 >

61C2
61C4

Character table of C61⋊C4

 class 124A4B61A61B61C61D61E61F61G61H61I61J61K61L61M61N61O
 size 1616161444444444444444
ρ11111111111111111111    trivial
ρ211-1-1111111111111111    linear of order 2
ρ31-1i-i111111111111111    linear of order 4
ρ41-1-ii111111111111111    linear of order 4
ρ54000ζ6151614961126110ζ6146614361186115ζ6148614061216113ζ6141613761246120ζ615361346127618ζ6136613161306125ζ615961396122612ζ6142613561266119ζ615461456116617ζ615761446117614ζ61606150611161ζ6147613261296114ζ615861336128613ζ61566155616615ζ615261386123619    orthogonal faithful
ρ64000ζ61566155616615ζ615261386123619ζ6141613761246120ζ6151614961126110ζ615761446117614ζ6146614361186115ζ61606150611161ζ6148614061216113ζ615361346127618ζ615961396122612ζ6136613161306125ζ615461456116617ζ6147613261296114ζ615861336128613ζ6142613561266119    orthogonal faithful
ρ74000ζ615861336128613ζ6142613561266119ζ6151614961126110ζ61566155616615ζ615961396122612ζ615261386123619ζ6136613161306125ζ6141613761246120ζ615761446117614ζ61606150611161ζ6146614361186115ζ615361346127618ζ615461456116617ζ6147613261296114ζ6148614061216113    orthogonal faithful
ρ84000ζ615361346127618ζ6151614961126110ζ6147613261296114ζ615461456116617ζ6146614361186115ζ6141613761246120ζ6142613561266119ζ615861336128613ζ6136613161306125ζ615261386123619ζ6148614061216113ζ61606150611161ζ615961396122612ζ615761446117614ζ61566155616615    orthogonal faithful
ρ94000ζ6141613761246120ζ6136613161306125ζ6142613561266119ζ6148614061216113ζ615461456116617ζ61606150611161ζ615761446117614ζ615261386123619ζ6147613261296114ζ615361346127618ζ615961396122612ζ615861336128613ζ61566155616615ζ6151614961126110ζ6146614361186115    orthogonal faithful
ρ104000ζ615461456116617ζ6141613761246120ζ615861336128613ζ6147613261296114ζ6136613161306125ζ6148614061216113ζ615261386123619ζ61566155616615ζ61606150611161ζ6146614361186115ζ6142613561266119ζ615961396122612ζ615761446117614ζ615361346127618ζ6151614961126110    orthogonal faithful
ρ114000ζ6136613161306125ζ615461456116617ζ615961396122612ζ61606150611161ζ6141613761246120ζ6147613261296114ζ61566155616615ζ615761446117614ζ6148614061216113ζ6151614961126110ζ615861336128613ζ6142613561266119ζ615261386123619ζ6146614361186115ζ615361346127618    orthogonal faithful
ρ124000ζ6147613261296114ζ6148614061216113ζ61566155616615ζ615861336128613ζ61606150611161ζ6142613561266119ζ6146614361186115ζ6151614961126110ζ615961396122612ζ6136613161306125ζ615261386123619ζ615761446117614ζ615361346127618ζ615461456116617ζ6141613761246120    orthogonal faithful
ρ134000ζ6142613561266119ζ615961396122612ζ6146614361186115ζ615261386123619ζ615861336128613ζ615761446117614ζ615461456116617ζ6136613161306125ζ61566155616615ζ6147613261296114ζ615361346127618ζ6151614961126110ζ6141613761246120ζ6148614061216113ζ61606150611161    orthogonal faithful
ρ144000ζ615261386123619ζ615761446117614ζ6136613161306125ζ6146614361186115ζ61566155616615ζ615361346127618ζ6147613261296114ζ61606150611161ζ6151614961126110ζ615861336128613ζ615461456116617ζ6141613761246120ζ6148614061216113ζ6142613561266119ζ615961396122612    orthogonal faithful
ρ154000ζ6146614361186115ζ615361346127618ζ61606150611161ζ6136613161306125ζ6151614961126110ζ615461456116617ζ615861336128613ζ615961396122612ζ6141613761246120ζ61566155616615ζ6147613261296114ζ6148614061216113ζ6142613561266119ζ615261386123619ζ615761446117614    orthogonal faithful
ρ164000ζ615761446117614ζ61566155616615ζ615461456116617ζ615361346127618ζ615261386123619ζ6151614961126110ζ6148614061216113ζ6147613261296114ζ6146614361186115ζ6142613561266119ζ6141613761246120ζ6136613161306125ζ61606150611161ζ615961396122612ζ615861336128613    orthogonal faithful
ρ174000ζ61606150611161ζ6147613261296114ζ615761446117614ζ615961396122612ζ6148614061216113ζ615861336128613ζ6151614961126110ζ615361346127618ζ6142613561266119ζ6141613761246120ζ61566155616615ζ615261386123619ζ6146614361186115ζ6136613161306125ζ615461456116617    orthogonal faithful
ρ184000ζ615961396122612ζ615861336128613ζ615361346127618ζ615761446117614ζ6142613561266119ζ61566155616615ζ6141613761246120ζ615461456116617ζ615261386123619ζ6148614061216113ζ6151614961126110ζ6146614361186115ζ6136613161306125ζ61606150611161ζ6147613261296114    orthogonal faithful
ρ194000ζ6148614061216113ζ61606150611161ζ615261386123619ζ6142613561266119ζ6147613261296114ζ615961396122612ζ615361346127618ζ6146614361186115ζ615861336128613ζ615461456116617ζ615761446117614ζ61566155616615ζ6151614961126110ζ6141613761246120ζ6136613161306125    orthogonal faithful

Smallest permutation representation of C61⋊C4
On 61 points: primitive
Generators in S61
(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)
(2 12 61 51)(3 23 60 40)(4 34 59 29)(5 45 58 18)(6 56 57 7)(8 17 55 46)(9 28 54 35)(10 39 53 24)(11 50 52 13)(14 22 49 41)(15 33 48 30)(16 44 47 19)(20 27 43 36)(21 38 42 25)(26 32 37 31)

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

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

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

C61⋊C4 is a maximal quotient of   C61⋊C8

Matrix representation of C61⋊C4 in GL4(𝔽733) generated by

92100
667010
558001
72375713376
,
57750158375
45042945172
430427567305
65352388626
G:=sub<GL(4,GF(733))| [92,667,558,72,1,0,0,375,0,1,0,713,0,0,1,376],[577,450,430,653,501,429,427,52,583,45,567,388,75,172,305,626] >;

C61⋊C4 in GAP, Magma, Sage, TeX

C_{61}\rtimes C_4
% in TeX

G:=Group("C61:C4");
// GroupNames label

G:=SmallGroup(244,3);
// by ID

G=gap.SmallGroup(244,3);
# by ID

G:=PCGroup([3,-2,-2,-61,6,398,1085]);
// Polycyclic

G:=Group<a,b|a^61=b^4=1,b*a*b^-1=a^50>;
// generators/relations

Export

Subgroup lattice of C61⋊C4 in TeX
Character table of C61⋊C4 in TeX

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