direct product, p-group, abelian, monomial
Aliases: C2×C4×C16, SmallGroup(128,837)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C2×C4×C16 |
| C1 — C2×C4×C16 |
| C1 — C2×C4×C16 |
Generators and relations for C2×C4×C16
G = < a,b,c | a2=b4=c16=1, ab=ba, ac=ca, bc=cb >
Subgroups: 108, all normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C16, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, C2×C42, C22×C8, C4×C16, C2×C4×C8, C22×C16, C2×C4×C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C16, C42, C2×C8, C22×C4, C4×C8, C2×C16, C2×C42, C22×C8, C4×C16, C2×C4×C8, C22×C16, C2×C4×C16
►Regular action on 128 points
Generators in S128
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 118)(18 119)(19 120)(20 121)(21 122)(22 123)(23 124)(24 125)(25 126)(26 127)(27 128)(28 113)(29 114)(30 115)(31 116)(32 117)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)
(1 127 96 107)(2 128 81 108)(3 113 82 109)(4 114 83 110)(5 115 84 111)(6 116 85 112)(7 117 86 97)(8 118 87 98)(9 119 88 99)(10 120 89 100)(11 121 90 101)(12 122 91 102)(13 123 92 103)(14 124 93 104)(15 125 94 105)(16 126 95 106)(17 77 56 48)(18 78 57 33)(19 79 58 34)(20 80 59 35)(21 65 60 36)(22 66 61 37)(23 67 62 38)(24 68 63 39)(25 69 64 40)(26 70 49 41)(27 71 50 42)(28 72 51 43)(29 73 52 44)(30 74 53 45)(31 75 54 46)(32 76 55 47)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)
G:=sub<Sym(128)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,118)(18,119)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,113)(29,114)(30,115)(31,116)(32,117)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90), (1,127,96,107)(2,128,81,108)(3,113,82,109)(4,114,83,110)(5,115,84,111)(6,116,85,112)(7,117,86,97)(8,118,87,98)(9,119,88,99)(10,120,89,100)(11,121,90,101)(12,122,91,102)(13,123,92,103)(14,124,93,104)(15,125,94,105)(16,126,95,106)(17,77,56,48)(18,78,57,33)(19,79,58,34)(20,80,59,35)(21,65,60,36)(22,66,61,37)(23,67,62,38)(24,68,63,39)(25,69,64,40)(26,70,49,41)(27,71,50,42)(28,72,51,43)(29,73,52,44)(30,74,53,45)(31,75,54,46)(32,76,55,47), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,118)(18,119)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,113)(29,114)(30,115)(31,116)(32,117)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90), (1,127,96,107)(2,128,81,108)(3,113,82,109)(4,114,83,110)(5,115,84,111)(6,116,85,112)(7,117,86,97)(8,118,87,98)(9,119,88,99)(10,120,89,100)(11,121,90,101)(12,122,91,102)(13,123,92,103)(14,124,93,104)(15,125,94,105)(16,126,95,106)(17,77,56,48)(18,78,57,33)(19,79,58,34)(20,80,59,35)(21,65,60,36)(22,66,61,37)(23,67,62,38)(24,68,63,39)(25,69,64,40)(26,70,49,41)(27,71,50,42)(28,72,51,43)(29,73,52,44)(30,74,53,45)(31,75,54,46)(32,76,55,47), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128) );
G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,118),(18,119),(19,120),(20,121),(21,122),(22,123),(23,124),(24,125),(25,126),(26,127),(27,128),(28,113),(29,114),(30,115),(31,116),(32,117),(49,107),(50,108),(51,109),(52,110),(53,111),(54,112),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90)], [(1,127,96,107),(2,128,81,108),(3,113,82,109),(4,114,83,110),(5,115,84,111),(6,116,85,112),(7,117,86,97),(8,118,87,98),(9,119,88,99),(10,120,89,100),(11,121,90,101),(12,122,91,102),(13,123,92,103),(14,124,93,104),(15,125,94,105),(16,126,95,106),(17,77,56,48),(18,78,57,33),(19,79,58,34),(20,80,59,35),(21,65,60,36),(22,66,61,37),(23,67,62,38),(24,68,63,39),(25,69,64,40),(26,70,49,41),(27,71,50,42),(28,72,51,43),(29,73,52,44),(30,74,53,45),(31,75,54,46),(32,76,55,47)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)]])
128 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 8A | ··· | 8AF | 16A | ··· | 16BL |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | C8 | C16 |
| kernel | C2×C4×C16 | C4×C16 | C2×C4×C8 | C22×C16 | C4×C8 | C2×C16 | C2×C42 | C22×C8 | C42 | C2×C8 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 2 | 2 | 8 | 16 | 8 | 64 |
Matrix representation of C2×C4×C16 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 16 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 16 |
| 4 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 7 |
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[16,0,0,0,13,0,0,0,16],[4,0,0,0,16,0,0,0,7] >;
C2×C4×C16 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_{16} % in TeX
G:=Group("C2xC4xC16"); // GroupNames label
G:=SmallGroup(128,837);
// by ID
G=gap.SmallGroup(128,837);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,120,136,124]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^16=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations