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## G = C2×C31⋊C5order 310 = 2·5·31

### Direct product of C2 and C31⋊C5

Aliases: C2×C31⋊C5, C62⋊C5, C312C10, SmallGroup(310,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — C2×C31⋊C5
 Chief series C1 — C31 — C31⋊C5 — C2×C31⋊C5
 Lower central C31 — C2×C31⋊C5
 Upper central C1 — C2

Generators and relations for C2×C31⋊C5
G = < a,b,c | a2=b31=c5=1, ab=ba, ac=ca, cbc-1=b2 >

Character table of C2×C31⋊C5

 class 1 2 5A 5B 5C 5D 10A 10B 10C 10D 31A 31B 31C 31D 31E 31F 62A 62B 62C 62D 62E 62F size 1 1 31 31 31 31 31 31 31 31 5 5 5 5 5 5 5 5 5 5 5 5 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 ζ53 ζ52 ζ54 ζ5 -ζ53 -ζ52 -ζ54 -ζ5 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 10 ρ4 1 1 ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 ζ54 ζ5 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 5 ρ5 1 -1 ζ52 ζ53 ζ5 ζ54 -ζ52 -ζ53 -ζ5 -ζ54 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 10 ρ6 1 -1 ζ5 ζ54 ζ53 ζ52 -ζ5 -ζ54 -ζ53 -ζ52 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 10 ρ7 1 1 ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 ζ52 ζ53 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 5 ρ8 1 1 ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 ζ5 ζ54 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 5 ρ9 1 1 ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 ζ53 ζ52 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 5 ρ10 1 -1 ζ54 ζ5 ζ52 ζ53 -ζ54 -ζ5 -ζ52 -ζ53 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 10 ρ11 5 5 0 0 0 0 0 0 0 0 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 complex lifted from C31⋊C5 ρ12 5 -5 0 0 0 0 0 0 0 0 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3124+ζ3117+ζ3112+ζ316+ζ313 -ζ3130-ζ3129-ζ3127-ζ3123-ζ3115 -ζ3124-ζ3117-ζ3112-ζ316-ζ313 -ζ3128-ζ3125-ζ3119-ζ3114-ζ317 -ζ3126-ζ3122-ζ3121-ζ3113-ζ3111 -ζ3116-ζ318-ζ314-ζ312-ζ31 -ζ3120-ζ3118-ζ3110-ζ319-ζ315 complex faithful ρ13 5 -5 0 0 0 0 0 0 0 0 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 -ζ3116-ζ318-ζ314-ζ312-ζ31 -ζ3128-ζ3125-ζ3119-ζ3114-ζ317 -ζ3124-ζ3117-ζ3112-ζ316-ζ313 -ζ3120-ζ3118-ζ3110-ζ319-ζ315 -ζ3130-ζ3129-ζ3127-ζ3123-ζ3115 -ζ3126-ζ3122-ζ3121-ζ3113-ζ3111 complex faithful ρ14 5 -5 0 0 0 0 0 0 0 0 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3120+ζ3118+ζ3110+ζ319+ζ315 -ζ3128-ζ3125-ζ3119-ζ3114-ζ317 -ζ3120-ζ3118-ζ3110-ζ319-ζ315 -ζ3126-ζ3122-ζ3121-ζ3113-ζ3111 -ζ3116-ζ318-ζ314-ζ312-ζ31 -ζ3124-ζ3117-ζ3112-ζ316-ζ313 -ζ3130-ζ3129-ζ3127-ζ3123-ζ3115 complex faithful ρ15 5 5 0 0 0 0 0 0 0 0 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3116+ζ318+ζ314+ζ312+ζ31 complex lifted from C31⋊C5 ρ16 5 5 0 0 0 0 0 0 0 0 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 complex lifted from C31⋊C5 ρ17 5 -5 0 0 0 0 0 0 0 0 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 -ζ3126-ζ3122-ζ3121-ζ3113-ζ3111 -ζ3130-ζ3129-ζ3127-ζ3123-ζ3115 -ζ3116-ζ318-ζ314-ζ312-ζ31 -ζ3124-ζ3117-ζ3112-ζ316-ζ313 -ζ3120-ζ3118-ζ3110-ζ319-ζ315 -ζ3128-ζ3125-ζ3119-ζ3114-ζ317 complex faithful ρ18 5 5 0 0 0 0 0 0 0 0 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3124+ζ3117+ζ3112+ζ316+ζ313 complex lifted from C31⋊C5 ρ19 5 -5 0 0 0 0 0 0 0 0 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 -ζ3124-ζ3117-ζ3112-ζ316-ζ313 -ζ3126-ζ3122-ζ3121-ζ3113-ζ3111 -ζ3120-ζ3118-ζ3110-ζ319-ζ315 -ζ3130-ζ3129-ζ3127-ζ3123-ζ3115 -ζ3128-ζ3125-ζ3119-ζ3114-ζ317 -ζ3116-ζ318-ζ314-ζ312-ζ31 complex faithful ρ20 5 5 0 0 0 0 0 0 0 0 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3120+ζ3118+ζ3110+ζ319+ζ315 complex lifted from C31⋊C5 ρ21 5 5 0 0 0 0 0 0 0 0 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 complex lifted from C31⋊C5 ρ22 5 -5 0 0 0 0 0 0 0 0 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3116+ζ318+ζ314+ζ312+ζ31 -ζ3120-ζ3118-ζ3110-ζ319-ζ315 -ζ3116-ζ318-ζ314-ζ312-ζ31 -ζ3130-ζ3129-ζ3127-ζ3123-ζ3115 -ζ3128-ζ3125-ζ3119-ζ3114-ζ317 -ζ3126-ζ3122-ζ3121-ζ3113-ζ3111 -ζ3124-ζ3117-ζ3112-ζ316-ζ313 complex faithful

Smallest permutation representation of C2×C31⋊C5
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 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)(33 48 40 36 34)(35 49 56 44 38)(37 50 41 52 42)(39 51 57 60 46)(43 53 58 45 54)(47 55 59 61 62)

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

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

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

Matrix representation of C2×C31⋊C5 in GL6(𝔽311)

 310 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 35 27 88 97 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 52 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 214 286 96 112 302 0 52 37 58 307 198

G:=sub<GL(6,GF(311))| [310,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,35,1,0,0,0,0,27,0,1,0,0,0,88,0,0,1,0,0,97,0,0,0,1,0,1,0,0,0,0],[52,0,0,0,0,0,0,1,0,0,214,52,0,0,0,0,286,37,0,0,1,0,96,58,0,0,0,0,112,307,0,0,0,1,302,198] >;

C2×C31⋊C5 in GAP, Magma, Sage, TeX

C_2\times C_{31}\rtimes C_5
% in TeX

G:=Group("C2xC31:C5");
// GroupNames label

G:=SmallGroup(310,2);
// by ID

G=gap.SmallGroup(310,2);
# by ID

G:=PCGroup([3,-2,-5,-31,725]);
// Polycyclic

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

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