metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C68.3C4, D17⋊2C8, D34.3C4, Dic17.4C22, C17⋊2(C2×C8), C17⋊2C8⋊3C2, C4.3(C17⋊C4), C34.1(C2×C4), (C4×D17).5C2, C2.1(C2×C17⋊C4), SmallGroup(272,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C68.C4 |
Generators and relations for C68.C4
G = < a,b,c | a17=b2=c8=1, bab=a-1, cac-1=a4, cbc-1=a3b >
Smallest permutation representation of C68.C4
►On 136 points
Generators in S136
(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)(120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 20)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(84 85)(86 96)(87 95)(88 94)(89 93)(90 92)(97 102)(98 101)(99 100)(103 107)(104 106)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)
(1 120 52 100 28 114 35 85)(2 133 68 87 29 110 51 72)(3 129 67 91 30 106 50 76)(4 125 66 95 31 119 49 80)(5 121 65 99 32 115 48 84)(6 134 64 86 33 111 47 71)(7 130 63 90 34 107 46 75)(8 126 62 94 18 103 45 79)(9 122 61 98 19 116 44 83)(10 135 60 102 20 112 43 70)(11 131 59 89 21 108 42 74)(12 127 58 93 22 104 41 78)(13 123 57 97 23 117 40 82)(14 136 56 101 24 113 39 69)(15 132 55 88 25 109 38 73)(16 128 54 92 26 105 37 77)(17 124 53 96 27 118 36 81)
G:=sub<Sym(136)| (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)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,20)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(84,85)(86,96)(87,95)(88,94)(89,93)(90,92)(97,102)(98,101)(99,100)(103,107)(104,106)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129), (1,120,52,100,28,114,35,85)(2,133,68,87,29,110,51,72)(3,129,67,91,30,106,50,76)(4,125,66,95,31,119,49,80)(5,121,65,99,32,115,48,84)(6,134,64,86,33,111,47,71)(7,130,63,90,34,107,46,75)(8,126,62,94,18,103,45,79)(9,122,61,98,19,116,44,83)(10,135,60,102,20,112,43,70)(11,131,59,89,21,108,42,74)(12,127,58,93,22,104,41,78)(13,123,57,97,23,117,40,82)(14,136,56,101,24,113,39,69)(15,132,55,88,25,109,38,73)(16,128,54,92,26,105,37,77)(17,124,53,96,27,118,36,81)>;
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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,20)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(84,85)(86,96)(87,95)(88,94)(89,93)(90,92)(97,102)(98,101)(99,100)(103,107)(104,106)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129), (1,120,52,100,28,114,35,85)(2,133,68,87,29,110,51,72)(3,129,67,91,30,106,50,76)(4,125,66,95,31,119,49,80)(5,121,65,99,32,115,48,84)(6,134,64,86,33,111,47,71)(7,130,63,90,34,107,46,75)(8,126,62,94,18,103,45,79)(9,122,61,98,19,116,44,83)(10,135,60,102,20,112,43,70)(11,131,59,89,21,108,42,74)(12,127,58,93,22,104,41,78)(13,123,57,97,23,117,40,82)(14,136,56,101,24,113,39,69)(15,132,55,88,25,109,38,73)(16,128,54,92,26,105,37,77)(17,124,53,96,27,118,36,81) );
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),(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119),(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,20),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(84,85),(86,96),(87,95),(88,94),(89,93),(90,92),(97,102),(98,101),(99,100),(103,107),(104,106),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114),(120,136),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129)], [(1,120,52,100,28,114,35,85),(2,133,68,87,29,110,51,72),(3,129,67,91,30,106,50,76),(4,125,66,95,31,119,49,80),(5,121,65,99,32,115,48,84),(6,134,64,86,33,111,47,71),(7,130,63,90,34,107,46,75),(8,126,62,94,18,103,45,79),(9,122,61,98,19,116,44,83),(10,135,60,102,20,112,43,70),(11,131,59,89,21,108,42,74),(12,127,58,93,22,104,41,78),(13,123,57,97,23,117,40,82),(14,136,56,101,24,113,39,69),(15,132,55,88,25,109,38,73),(16,128,54,92,26,105,37,77),(17,124,53,96,27,118,36,81)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 68A | ··· | 68H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 17 | 17 | 17 | 17 | 34 | 34 | 34 | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 17 | 17 | 17 | ··· | 17 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C17⋊C4 | C2×C17⋊C4 | C68.C4 |
| kernel | C68.C4 | C17⋊2C8 | C4×D17 | C68 | D34 | D17 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 4 | 4 | 8 |
Matrix representation of C68.C4 ►in GL4(𝔽137) generated by
| 136 | 1 | 0 | 0 |
| 136 | 0 | 1 | 0 |
| 136 | 0 | 0 | 1 |
| 85 | 104 | 33 | 51 |
| 136 | 0 | 0 | 0 |
| 32 | 104 | 86 | 1 |
| 5 | 47 | 3 | 51 |
| 110 | 76 | 114 | 31 |
| 10 | 0 | 127 | 35 |
| 84 | 98 | 14 | 79 |
| 106 | 76 | 131 | 79 |
| 63 | 84 | 12 | 35 |
G:=sub<GL(4,GF(137))| [136,136,136,85,1,0,0,104,0,1,0,33,0,0,1,51],[136,32,5,110,0,104,47,76,0,86,3,114,0,1,51,31],[10,84,106,63,0,98,76,84,127,14,131,12,35,79,79,35] >;
C68.C4 in GAP, Magma, Sage, TeX
C_{68}.C_4 % in TeX
G:=Group("C68.C4"); // GroupNames label
G:=SmallGroup(272,29);
// by ID
G=gap.SmallGroup(272,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,46,42,5204,1614]);
// Polycyclic
G:=Group<a,b,c|a^17=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^4,c*b*c^-1=a^3*b>;
// generators/relations
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