direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C22, C10⋊C22, C110⋊3C2, C55⋊4C22, C5⋊(C2×C22), SmallGroup(220,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C22 |
Generators and relations for D5×C22
G = < a,b,c | a22=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C22
►On 110 points
Generators in S110
(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)
(1 65 78 43 107)(2 66 79 44 108)(3 45 80 23 109)(4 46 81 24 110)(5 47 82 25 89)(6 48 83 26 90)(7 49 84 27 91)(8 50 85 28 92)(9 51 86 29 93)(10 52 87 30 94)(11 53 88 31 95)(12 54 67 32 96)(13 55 68 33 97)(14 56 69 34 98)(15 57 70 35 99)(16 58 71 36 100)(17 59 72 37 101)(18 60 73 38 102)(19 61 74 39 103)(20 62 75 40 104)(21 63 76 41 105)(22 64 77 42 106)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)
G:=sub<Sym(110)| (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), (1,65,78,43,107)(2,66,79,44,108)(3,45,80,23,109)(4,46,81,24,110)(5,47,82,25,89)(6,48,83,26,90)(7,49,84,27,91)(8,50,85,28,92)(9,51,86,29,93)(10,52,87,30,94)(11,53,88,31,95)(12,54,67,32,96)(13,55,68,33,97)(14,56,69,34,98)(15,57,70,35,99)(16,58,71,36,100)(17,59,72,37,101)(18,60,73,38,102)(19,61,74,39,103)(20,62,75,40,104)(21,63,76,41,105)(22,64,77,42,106), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)>;
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), (1,65,78,43,107)(2,66,79,44,108)(3,45,80,23,109)(4,46,81,24,110)(5,47,82,25,89)(6,48,83,26,90)(7,49,84,27,91)(8,50,85,28,92)(9,51,86,29,93)(10,52,87,30,94)(11,53,88,31,95)(12,54,67,32,96)(13,55,68,33,97)(14,56,69,34,98)(15,57,70,35,99)(16,58,71,36,100)(17,59,72,37,101)(18,60,73,38,102)(19,61,74,39,103)(20,62,75,40,104)(21,63,76,41,105)(22,64,77,42,106), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88) );
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)], [(1,65,78,43,107),(2,66,79,44,108),(3,45,80,23,109),(4,46,81,24,110),(5,47,82,25,89),(6,48,83,26,90),(7,49,84,27,91),(8,50,85,28,92),(9,51,86,29,93),(10,52,87,30,94),(11,53,88,31,95),(12,54,67,32,96),(13,55,68,33,97),(14,56,69,34,98),(15,57,70,35,99),(16,58,71,36,100),(17,59,72,37,101),(18,60,73,38,102),(19,61,74,39,103),(20,62,75,40,104),(21,63,76,41,105),(22,64,77,42,106)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,45),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88)]])
D5×C22 is a maximal subgroup of
C55⋊D4 C11⋊D20
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AD | 55A | ··· | 55T | 110A | ··· | 110T |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D5 | D10 | D5×C11 | D5×C22 |
| kernel | D5×C22 | D5×C11 | C110 | D10 | D5 | C10 | C22 | C11 | C2 | C1 |
| # reps | 1 | 2 | 1 | 10 | 20 | 10 | 2 | 2 | 20 | 20 |
Matrix representation of D5×C22 ►in GL3(𝔽331) generated by
| 330 | 0 | 0 |
| 0 | 111 | 0 |
| 0 | 0 | 111 |
| 1 | 0 | 0 |
| 0 | 215 | 216 |
| 0 | 330 | 330 |
| 1 | 0 | 0 |
| 0 | 330 | 115 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(331))| [330,0,0,0,111,0,0,0,111],[1,0,0,0,215,330,0,216,330],[1,0,0,0,330,0,0,115,1] >;
D5×C22 in GAP, Magma, Sage, TeX
D_5\times C_{22} % in TeX
G:=Group("D5xC22"); // GroupNames label
G:=SmallGroup(220,13);
// by ID
G=gap.SmallGroup(220,13);
# by ID
G:=PCGroup([4,-2,-2,-11,-5,2819]);
// Polycyclic
G:=Group<a,b,c|a^22=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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