direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2xC53:C4, C106:C4, D53:C4, D106.C2, D53.C22, C53:(C2xC4), SmallGroup(424,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C2xC53:C4 |
Generators and relations for C2xC53:C4
G = < a,b,c | a2=b53=c4=1, ab=ba, ac=ca, cbc-1=b23 >
Quotients: C1, C2, C4, C22, C2xC4, C53:C4, C2xC53:C4
Smallest permutation representation of C2xC53:C4
►On 106 points
Generators in S106
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)
(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)
(1 54)(2 84 53 77)(3 61 52 100)(4 91 51 70)(5 68 50 93)(6 98 49 63)(7 75 48 86)(8 105 47 56)(9 82 46 79)(10 59 45 102)(11 89 44 72)(12 66 43 95)(13 96 42 65)(14 73 41 88)(15 103 40 58)(16 80 39 81)(17 57 38 104)(18 87 37 74)(19 64 36 97)(20 94 35 67)(21 71 34 90)(22 101 33 60)(23 78 32 83)(24 55 31 106)(25 85 30 76)(26 62 29 99)(27 92 28 69)
G:=sub<Sym(106)| (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106), (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), (1,54)(2,84,53,77)(3,61,52,100)(4,91,51,70)(5,68,50,93)(6,98,49,63)(7,75,48,86)(8,105,47,56)(9,82,46,79)(10,59,45,102)(11,89,44,72)(12,66,43,95)(13,96,42,65)(14,73,41,88)(15,103,40,58)(16,80,39,81)(17,57,38,104)(18,87,37,74)(19,64,36,97)(20,94,35,67)(21,71,34,90)(22,101,33,60)(23,78,32,83)(24,55,31,106)(25,85,30,76)(26,62,29,99)(27,92,28,69)>;
G:=Group( (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106), (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), (1,54)(2,84,53,77)(3,61,52,100)(4,91,51,70)(5,68,50,93)(6,98,49,63)(7,75,48,86)(8,105,47,56)(9,82,46,79)(10,59,45,102)(11,89,44,72)(12,66,43,95)(13,96,42,65)(14,73,41,88)(15,103,40,58)(16,80,39,81)(17,57,38,104)(18,87,37,74)(19,64,36,97)(20,94,35,67)(21,71,34,90)(22,101,33,60)(23,78,32,83)(24,55,31,106)(25,85,30,76)(26,62,29,99)(27,92,28,69) );
G=PermutationGroup([[(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(46,99),(47,100),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106)], [(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)], [(1,54),(2,84,53,77),(3,61,52,100),(4,91,51,70),(5,68,50,93),(6,98,49,63),(7,75,48,86),(8,105,47,56),(9,82,46,79),(10,59,45,102),(11,89,44,72),(12,66,43,95),(13,96,42,65),(14,73,41,88),(15,103,40,58),(16,80,39,81),(17,57,38,104),(18,87,37,74),(19,64,36,97),(20,94,35,67),(21,71,34,90),(22,101,33,60),(23,78,32,83),(24,55,31,106),(25,85,30,76),(26,62,29,99),(27,92,28,69)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 53A | ··· | 53M | 106A | ··· | 106M |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 53 | ··· | 53 | 106 | ··· | 106 |
| size | 1 | 1 | 53 | 53 | 53 | 53 | 53 | 53 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C4 | C4 | C53:C4 | C2xC53:C4 |
| kernel | C2xC53:C4 | C53:C4 | D106 | D53 | C106 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 13 | 13 |
Matrix representation of C2xC53:C4 ►in GL4(F1061) generated by
| 1060 | 0 | 0 | 0 |
| 0 | 1060 | 0 | 0 |
| 0 | 0 | 1060 | 0 |
| 0 | 0 | 0 | 1060 |
| 80 | 1043 | 80 | 1060 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1060 | 0 | 0 | 0 |
| 221 | 540 | 590 | 637 |
| 98 | 712 | 552 | 692 |
| 820 | 52 | 98 | 1031 |
G:=sub<GL(4,GF(1061))| [1060,0,0,0,0,1060,0,0,0,0,1060,0,0,0,0,1060],[80,1,0,0,1043,0,1,0,80,0,0,1,1060,0,0,0],[1060,221,98,820,0,540,712,52,0,590,552,98,0,637,692,1031] >;
C2xC53:C4 in GAP, Magma, Sage, TeX
C_2\times C_{53}\rtimes C_4 % in TeX
G:=Group("C2xC53:C4"); // GroupNames label
G:=SmallGroup(424,12);
// by ID
G=gap.SmallGroup(424,12);
# by ID
G:=PCGroup([4,-2,-2,-2,-53,16,3843,1675]);
// Polycyclic
G:=Group<a,b,c|a^2=b^53=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^23>;
// generators/relations
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