direct product, p-group, abelian, monomial
Aliases: C2×C64, SmallGroup(128,159)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C2×C64 |
| C1 — C2×C64 |
| C1 — C2×C64 |
| C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C8 — C8 — C8 — C8 — C16 — C16 — C32 — C2×C64 |
Generators and relations for C2×C64
G = < a,b | a2=b64=1, ab=ba >
Smallest permutation representation of C2×C64
►Regular action on 128 points
Generators in S128
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 81)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 89)
(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)| (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,89), (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( (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,89), (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([[(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,81),(57,82),(58,83),(59,84),(60,85),(61,86),(62,87),(63,88),(64,89)], [(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 | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 16A | ··· | 16P | 32A | ··· | 32AF | 64A | ··· | 64BL |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 | 32 | ··· | 32 | 64 | ··· | 64 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | C32 | C64 |
| kernel | C2×C64 | C64 | C2×C32 | C32 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 16 | 64 |
Matrix representation of C2×C64 ►in GL2(𝔽193) generated by
| 1 | 0 |
| 0 | 192 |
| 119 | 0 |
| 0 | 42 |
G:=sub<GL(2,GF(193))| [1,0,0,192],[119,0,0,42] >;
C2×C64 in GAP, Magma, Sage, TeX
C_2\times C_{64} % in TeX
G:=Group("C2xC64"); // GroupNames label
G:=SmallGroup(128,159);
// by ID
G=gap.SmallGroup(128,159);
# by ID
G:=PCGroup([7,-2,2,-2,-2,-2,-2,-2,28,58,80,102,124]);
// Polycyclic
G:=Group<a,b|a^2=b^64=1,a*b=b*a>;
// generators/relations
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