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
(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