direct product, p-group, metabelian, nilpotent (class 5), monomial
Aliases: C2×Q64, C8.20D8, C4.8D16, C16.11D4, C16.8C23, C32.5C22, C22.16D16, Q32.1C22, (C2×C32).4C2, (C2×C4).90D8, C4.15(C2×D8), C8.47(C2×D4), C2.14(C2×D16), (C2×C8).259D4, (C2×Q32).4C2, (C2×C16).90C22, SmallGroup(128,993)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×Q64
G = < a,b,c | a2=b32=1, c2=b16, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Q64
►Regular action on 128 points
Generators in S128
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 121)(13 122)(14 123)(15 124)(16 125)(17 126)(18 127)(19 128)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 65)(62 66)(63 67)(64 68)
(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)
(1 92 17 76)(2 91 18 75)(3 90 19 74)(4 89 20 73)(5 88 21 72)(6 87 22 71)(7 86 23 70)(8 85 24 69)(9 84 25 68)(10 83 26 67)(11 82 27 66)(12 81 28 65)(13 80 29 96)(14 79 30 95)(15 78 31 94)(16 77 32 93)(33 117 49 101)(34 116 50 100)(35 115 51 99)(36 114 52 98)(37 113 53 97)(38 112 54 128)(39 111 55 127)(40 110 56 126)(41 109 57 125)(42 108 58 124)(43 107 59 123)(44 106 60 122)(45 105 61 121)(46 104 62 120)(47 103 63 119)(48 102 64 118)
G:=sub<Sym(128)| (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,121)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,65)(62,66)(63,67)(64,68), (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), (1,92,17,76)(2,91,18,75)(3,90,19,74)(4,89,20,73)(5,88,21,72)(6,87,22,71)(7,86,23,70)(8,85,24,69)(9,84,25,68)(10,83,26,67)(11,82,27,66)(12,81,28,65)(13,80,29,96)(14,79,30,95)(15,78,31,94)(16,77,32,93)(33,117,49,101)(34,116,50,100)(35,115,51,99)(36,114,52,98)(37,113,53,97)(38,112,54,128)(39,111,55,127)(40,110,56,126)(41,109,57,125)(42,108,58,124)(43,107,59,123)(44,106,60,122)(45,105,61,121)(46,104,62,120)(47,103,63,119)(48,102,64,118)>;
G:=Group( (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,121)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,65)(62,66)(63,67)(64,68), (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), (1,92,17,76)(2,91,18,75)(3,90,19,74)(4,89,20,73)(5,88,21,72)(6,87,22,71)(7,86,23,70)(8,85,24,69)(9,84,25,68)(10,83,26,67)(11,82,27,66)(12,81,28,65)(13,80,29,96)(14,79,30,95)(15,78,31,94)(16,77,32,93)(33,117,49,101)(34,116,50,100)(35,115,51,99)(36,114,52,98)(37,113,53,97)(38,112,54,128)(39,111,55,127)(40,110,56,126)(41,109,57,125)(42,108,58,124)(43,107,59,123)(44,106,60,122)(45,105,61,121)(46,104,62,120)(47,103,63,119)(48,102,64,118) );
G=PermutationGroup([[(1,110),(2,111),(3,112),(4,113),(5,114),(6,115),(7,116),(8,117),(9,118),(10,119),(11,120),(12,121),(13,122),(14,123),(15,124),(16,125),(17,126),(18,127),(19,128),(20,97),(21,98),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,106),(30,107),(31,108),(32,109),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,65),(62,66),(63,67),(64,68)], [(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)], [(1,92,17,76),(2,91,18,75),(3,90,19,74),(4,89,20,73),(5,88,21,72),(6,87,22,71),(7,86,23,70),(8,85,24,69),(9,84,25,68),(10,83,26,67),(11,82,27,66),(12,81,28,65),(13,80,29,96),(14,79,30,95),(15,78,31,94),(16,77,32,93),(33,117,49,101),(34,116,50,100),(35,115,51,99),(36,114,52,98),(37,113,53,97),(38,112,54,128),(39,111,55,127),(40,110,56,126),(41,109,57,125),(42,108,58,124),(43,107,59,123),(44,106,60,122),(45,105,61,121),(46,104,62,120),(47,103,63,119),(48,102,64,118)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | D16 | D16 | Q64 |
| kernel | C2×Q64 | C2×C32 | Q64 | C2×Q32 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of C2×Q64 ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 96 | 0 |
| 0 | 0 | 96 |
| 96 | 0 | 0 |
| 0 | 16 | 91 |
| 0 | 6 | 16 |
| 96 | 0 | 0 |
| 0 | 69 | 66 |
| 0 | 66 | 28 |
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[96,0,0,0,16,6,0,91,16],[96,0,0,0,69,66,0,66,28] >;
C2×Q64 in GAP, Magma, Sage, TeX
C_2\times Q_{64} % in TeX
G:=Group("C2xQ64"); // GroupNames label
G:=SmallGroup(128,993);
// by ID
G=gap.SmallGroup(128,993);
# by ID
G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,448,141,456,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^2=b^32=1,c^2=b^16,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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