p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2.1C82, C42.66Q8, C42.452D4, C23.39C42, (C2×C8)⋊3C8, C4.21(C4⋊C8), C2.1(C8⋊C8), C22.11(C4×C8), (C22×C8).21C4, C4.27(C22⋊C8), (C2×C4).85M4(2), C22.10(C8⋊C4), (C2×C42).1141C22, C2.1(C22.7C42), C22.14(C2.C42), (C2×C4×C8).1C2, (C2×C4).91(C2×C8), (C2×C4).154(C4⋊C4), (C22×C4).500(C2×C4), (C2×C4).367(C22⋊C4), SmallGroup(128,5)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.C82
G = < a,b,c | a2=b8=c8=1, cbc-1=ab=ba, ac=ca >
Subgroups: 136 in 106 conjugacy classes, 76 normal (7 characteristic)
C1, C2, C2 [×6], C4 [×12], C22, C22 [×6], C8 [×12], C2×C4 [×18], C23, C42, C42 [×3], C2×C8 [×12], C2×C8 [×12], C22×C4 [×3], C4×C8 [×6], C2×C42, C22×C8 [×6], C2×C4×C8 [×3], C2.C82
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×12], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×6], M4(2) [×6], C2.C42, C4×C8 [×3], C8⋊C4 [×3], C22⋊C8 [×6], C4⋊C8 [×6], C82, C8⋊C8 [×3], C22.7C42 [×3], C2.C82
►Regular action on 128 points
Generators in S128
(1 95)(2 96)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 126)(10 127)(11 128)(12 121)(13 122)(14 123)(15 124)(16 125)(17 103)(18 104)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 81)(32 82)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(105 113)(106 114)(107 115)(108 116)(109 117)(110 118)(111 119)(112 120)
(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 66 55 121 105 37 87 103)(2 75 56 13 106 46 88 18)(3 68 49 123 107 39 81 97)(4 77 50 15 108 48 82 20)(5 70 51 125 109 33 83 99)(6 79 52 9 110 42 84 22)(7 72 53 127 111 35 85 101)(8 73 54 11 112 44 86 24)(10 119 43 27 23 93 80 61)(12 113 45 29 17 95 74 63)(14 115 47 31 19 89 76 57)(16 117 41 25 21 91 78 59)(26 100 92 71 60 126 118 34)(28 102 94 65 62 128 120 36)(30 104 96 67 64 122 114 38)(32 98 90 69 58 124 116 40)
G:=sub<Sym(128)| (1,95)(2,96)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,103)(18,104)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,81)(32,82)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120), (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), (1,66,55,121,105,37,87,103)(2,75,56,13,106,46,88,18)(3,68,49,123,107,39,81,97)(4,77,50,15,108,48,82,20)(5,70,51,125,109,33,83,99)(6,79,52,9,110,42,84,22)(7,72,53,127,111,35,85,101)(8,73,54,11,112,44,86,24)(10,119,43,27,23,93,80,61)(12,113,45,29,17,95,74,63)(14,115,47,31,19,89,76,57)(16,117,41,25,21,91,78,59)(26,100,92,71,60,126,118,34)(28,102,94,65,62,128,120,36)(30,104,96,67,64,122,114,38)(32,98,90,69,58,124,116,40)>;
G:=Group( (1,95)(2,96)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,103)(18,104)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,81)(32,82)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120), (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), (1,66,55,121,105,37,87,103)(2,75,56,13,106,46,88,18)(3,68,49,123,107,39,81,97)(4,77,50,15,108,48,82,20)(5,70,51,125,109,33,83,99)(6,79,52,9,110,42,84,22)(7,72,53,127,111,35,85,101)(8,73,54,11,112,44,86,24)(10,119,43,27,23,93,80,61)(12,113,45,29,17,95,74,63)(14,115,47,31,19,89,76,57)(16,117,41,25,21,91,78,59)(26,100,92,71,60,126,118,34)(28,102,94,65,62,128,120,36)(30,104,96,67,64,122,114,38)(32,98,90,69,58,124,116,40) );
G=PermutationGroup([(1,95),(2,96),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,126),(10,127),(11,128),(12,121),(13,122),(14,123),(15,124),(16,125),(17,103),(18,104),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,81),(32,82),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(105,113),(106,114),(107,115),(108,116),(109,117),(110,118),(111,119),(112,120)], [(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)], [(1,66,55,121,105,37,87,103),(2,75,56,13,106,46,88,18),(3,68,49,123,107,39,81,97),(4,77,50,15,108,48,82,20),(5,70,51,125,109,33,83,99),(6,79,52,9,110,42,84,22),(7,72,53,127,111,35,85,101),(8,73,54,11,112,44,86,24),(10,119,43,27,23,93,80,61),(12,113,45,29,17,95,74,63),(14,115,47,31,19,89,76,57),(16,117,41,25,21,91,78,59),(26,100,92,71,60,126,118,34),(28,102,94,65,62,128,120,36),(30,104,96,67,64,122,114,38),(32,98,90,69,58,124,116,40)])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 8A | ··· | 8AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D4 | Q8 | M4(2) |
| kernel | C2.C82 | C2×C4×C8 | C22×C8 | C2×C8 | C42 | C42 | C2×C4 |
| # reps | 1 | 3 | 12 | 48 | 3 | 1 | 12 |
Matrix representation of C2.C82 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 10 | 1 |
| 0 | 0 | 1 | 7 |
| 8 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[8,0,0,0,0,8,0,0,0,0,10,1,0,0,1,7],[8,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0] >;
C2.C82 in GAP, Magma, Sage, TeX
C_2.C_8^2
% in TeX
G:=Group("C2.C8^2"); // GroupNames label
G:=SmallGroup(128,5);
// by ID
G=gap.SmallGroup(128,5);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,136,172]);
// Polycyclic
G:=Group<a,b,c|a^2=b^8=c^8=1,c*b*c^-1=a*b=b*a,a*c=c*a>;
// generators/relations