direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊1C8, C42.49Q8, C42.318D4, C42.620C23, (C2×C8)⋊5C8, C8⋊7(C2×C8), C4○(C8⋊1C8), (C4×C8).25C4, C4.82(C2×D8), C4.11(C4⋊C8), (C2×C4).165D8, (C2×C4).68Q16, C4.54(C2×Q16), (C22×C8).33C4, C4.25(C22×C8), C4.23(C2.D8), (C22×C4).73Q8, C4⋊C8.264C22, C23.80(C4⋊C4), C22.21(C4⋊C8), C42.308(C2×C4), (C4×C8).388C22, (C22×C4).810D4, C4.40(C2×M4(2)), (C2×C4).74M4(2), C22.20(C2.D8), C22.11(C8.C4), (C2×C42).1039C22, C2.5(C2×C4⋊C8), (C2×C4×C8).31C2, (C2×C4)○(C8⋊1C8), (C2×C4⋊C8).17C2, C2.1(C2×C2.D8), (C2×C4).81(C2×C8), (C2×C8).218(C2×C4), C2.2(C2×C8.C4), C22.46(C2×C4⋊C4), (C2×C4).147(C2×Q8), (C2×C4).115(C4⋊C4), (C2×C4).1456(C2×D4), (C2×C4).502(C22×C4), (C22×C4).472(C2×C4), SmallGroup(128,295)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊1C8
G = < a,b,c | a2=b8=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 140 in 100 conjugacy classes, 76 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C4×C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C22×C8, C8⋊1C8, C2×C4×C8, C2×C4⋊C8, C2×C8⋊1C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), D8, Q16, C22×C4, C2×D4, C2×Q8, C4⋊C8, C2.D8, C8.C4, C2×C4⋊C4, C22×C8, C2×M4(2), C2×D8, C2×Q16, C8⋊1C8, C2×C4⋊C8, C2×C2.D8, C2×C8.C4, C2×C8⋊1C8
►Regular action on 128 points
Generators in S128
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(25 86)(26 87)(27 88)(28 81)(29 82)(30 83)(31 84)(32 85)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 97)(48 98)(49 79)(50 80)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 65)(64 66)(105 124)(106 125)(107 126)(108 127)(109 128)(110 121)(111 122)(112 123)
(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 69 23 120 103 49 82 127)(2 68 24 119 104 56 83 126)(3 67 17 118 97 55 84 125)(4 66 18 117 98 54 85 124)(5 65 19 116 99 53 86 123)(6 72 20 115 100 52 87 122)(7 71 21 114 101 51 88 121)(8 70 22 113 102 50 81 128)(9 48 76 32 105 37 64 95)(10 47 77 31 106 36 57 94)(11 46 78 30 107 35 58 93)(12 45 79 29 108 34 59 92)(13 44 80 28 109 33 60 91)(14 43 73 27 110 40 61 90)(15 42 74 26 111 39 62 89)(16 41 75 25 112 38 63 96)
G:=sub<Sym(128)| (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(25,86)(26,87)(27,88)(28,81)(29,82)(30,83)(31,84)(32,85)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(49,79)(50,80)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,65)(64,66)(105,124)(106,125)(107,126)(108,127)(109,128)(110,121)(111,122)(112,123), (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), (1,69,23,120,103,49,82,127)(2,68,24,119,104,56,83,126)(3,67,17,118,97,55,84,125)(4,66,18,117,98,54,85,124)(5,65,19,116,99,53,86,123)(6,72,20,115,100,52,87,122)(7,71,21,114,101,51,88,121)(8,70,22,113,102,50,81,128)(9,48,76,32,105,37,64,95)(10,47,77,31,106,36,57,94)(11,46,78,30,107,35,58,93)(12,45,79,29,108,34,59,92)(13,44,80,28,109,33,60,91)(14,43,73,27,110,40,61,90)(15,42,74,26,111,39,62,89)(16,41,75,25,112,38,63,96)>;
G:=Group( (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(25,86)(26,87)(27,88)(28,81)(29,82)(30,83)(31,84)(32,85)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(49,79)(50,80)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,65)(64,66)(105,124)(106,125)(107,126)(108,127)(109,128)(110,121)(111,122)(112,123), (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), (1,69,23,120,103,49,82,127)(2,68,24,119,104,56,83,126)(3,67,17,118,97,55,84,125)(4,66,18,117,98,54,85,124)(5,65,19,116,99,53,86,123)(6,72,20,115,100,52,87,122)(7,71,21,114,101,51,88,121)(8,70,22,113,102,50,81,128)(9,48,76,32,105,37,64,95)(10,47,77,31,106,36,57,94)(11,46,78,30,107,35,58,93)(12,45,79,29,108,34,59,92)(13,44,80,28,109,33,60,91)(14,43,73,27,110,40,61,90)(15,42,74,26,111,39,62,89)(16,41,75,25,112,38,63,96) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,33),(9,117),(10,118),(11,119),(12,120),(13,113),(14,114),(15,115),(16,116),(17,94),(18,95),(19,96),(20,89),(21,90),(22,91),(23,92),(24,93),(25,86),(26,87),(27,88),(28,81),(29,82),(30,83),(31,84),(32,85),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,97),(48,98),(49,79),(50,80),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,65),(64,66),(105,124),(106,125),(107,126),(108,127),(109,128),(110,121),(111,122),(112,123)], [(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)], [(1,69,23,120,103,49,82,127),(2,68,24,119,104,56,83,126),(3,67,17,118,97,55,84,125),(4,66,18,117,98,54,85,124),(5,65,19,116,99,53,86,123),(6,72,20,115,100,52,87,122),(7,71,21,114,101,51,88,121),(8,70,22,113,102,50,81,128),(9,48,76,32,105,37,64,95),(10,47,77,31,106,36,57,94),(11,46,78,30,107,35,58,93),(12,45,79,29,108,34,59,92),(13,44,80,28,109,33,60,91),(14,43,73,27,110,40,61,90),(15,42,74,26,111,39,62,89),(16,41,75,25,112,38,63,96)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P | 8Q | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D4 | Q8 | M4(2) | D8 | Q16 | C8.C4 |
| kernel | C2×C8⋊1C8 | C8⋊1C8 | C2×C4×C8 | C2×C4⋊C8 | C4×C8 | C22×C8 | C2×C8 | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 |
Matrix representation of C2×C8⋊1C8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 2 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 5 | 7 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[2,0,0,0,0,16,0,0,0,0,10,5,0,0,7,7] >;
C2×C8⋊1C8 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_1C_8
% in TeX
G:=Group("C2xC8:1C8"); // GroupNames label
G:=SmallGroup(128,295);
// by ID
G=gap.SmallGroup(128,295);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,288,1123,136,172]);
// Polycyclic
G:=Group<a,b,c|a^2=b^8=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations