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## G = C2×C82order 128 = 27

### Abelian group of type [2,8,8]

Aliases: C2×C82, SmallGroup(128,179)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C82
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×C8 — C2×C82
 Lower central C1 — C2×C82
 Upper central C1 — C2×C82
 Jennings C1 — C22 — C22 — C42 — C2×C82

Generators and relations for C2×C82
G = < a,b,c | a2=b8=c8=1, ab=ba, ac=ca, bc=cb >

Subgroups: 140, all normal (6 characteristic)
C1, C2 [×7], C4 [×12], C22, C22 [×6], C8 [×24], C2×C4 [×18], C23, C42, C42 [×3], C2×C8 [×36], C22×C4 [×3], C4×C8 [×12], C2×C42, C22×C8 [×6], C82 [×4], C2×C4×C8 [×3], C2×C82
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×24], C2×C4 [×18], C23, C42 [×4], C2×C8 [×36], C22×C4 [×3], C4×C8 [×12], C2×C42, C22×C8 [×6], C82 [×4], C2×C4×C8 [×3], C2×C82

Smallest permutation representation of C2×C82
Regular action on 128 points
Generators in S128
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 126)(66 127)(67 128)(68 121)(69 122)(70 123)(71 124)(72 125)
(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 10 119 47 31 121 105 93)(2 11 120 48 32 122 106 94)(3 12 113 41 25 123 107 95)(4 13 114 42 26 124 108 96)(5 14 115 43 27 125 109 89)(6 15 116 44 28 126 110 90)(7 16 117 45 29 127 111 91)(8 9 118 46 30 128 112 92)(17 78 59 82 99 72 53 33)(18 79 60 83 100 65 54 34)(19 80 61 84 101 66 55 35)(20 73 62 85 102 67 56 36)(21 74 63 86 103 68 49 37)(22 75 64 87 104 69 50 38)(23 76 57 88 97 70 51 39)(24 77 58 81 98 71 52 40)

G:=sub<Sym(128)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,126)(66,127)(67,128)(68,121)(69,122)(70,123)(71,124)(72,125), (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), (1,10,119,47,31,121,105,93)(2,11,120,48,32,122,106,94)(3,12,113,41,25,123,107,95)(4,13,114,42,26,124,108,96)(5,14,115,43,27,125,109,89)(6,15,116,44,28,126,110,90)(7,16,117,45,29,127,111,91)(8,9,118,46,30,128,112,92)(17,78,59,82,99,72,53,33)(18,79,60,83,100,65,54,34)(19,80,61,84,101,66,55,35)(20,73,62,85,102,67,56,36)(21,74,63,86,103,68,49,37)(22,75,64,87,104,69,50,38)(23,76,57,88,97,70,51,39)(24,77,58,81,98,71,52,40)>;

G:=Group( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,126)(66,127)(67,128)(68,121)(69,122)(70,123)(71,124)(72,125), (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), (1,10,119,47,31,121,105,93)(2,11,120,48,32,122,106,94)(3,12,113,41,25,123,107,95)(4,13,114,42,26,124,108,96)(5,14,115,43,27,125,109,89)(6,15,116,44,28,126,110,90)(7,16,117,45,29,127,111,91)(8,9,118,46,30,128,112,92)(17,78,59,82,99,72,53,33)(18,79,60,83,100,65,54,34)(19,80,61,84,101,66,55,35)(20,73,62,85,102,67,56,36)(21,74,63,86,103,68,49,37)(22,75,64,87,104,69,50,38)(23,76,57,88,97,70,51,39)(24,77,58,81,98,71,52,40) );

G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,88),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,126),(66,127),(67,128),(68,121),(69,122),(70,123),(71,124),(72,125)], [(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)], [(1,10,119,47,31,121,105,93),(2,11,120,48,32,122,106,94),(3,12,113,41,25,123,107,95),(4,13,114,42,26,124,108,96),(5,14,115,43,27,125,109,89),(6,15,116,44,28,126,110,90),(7,16,117,45,29,127,111,91),(8,9,118,46,30,128,112,92),(17,78,59,82,99,72,53,33),(18,79,60,83,100,65,54,34),(19,80,61,84,101,66,55,35),(20,73,62,85,102,67,56,36),(21,74,63,86,103,68,49,37),(22,75,64,87,104,69,50,38),(23,76,57,88,97,70,51,39),(24,77,58,81,98,71,52,40)])

128 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 8A ··· 8CR order 1 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 1 ··· 1

128 irreducible representations

 dim 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C8 kernel C2×C82 C82 C2×C4×C8 C4×C8 C22×C8 C2×C8 # reps 1 4 3 12 12 96

Matrix representation of C2×C82 in GL3(𝔽17) generated by

 1 0 0 0 16 0 0 0 16
,
 13 0 0 0 13 0 0 0 8
,
 15 0 0 0 2 0 0 0 13
G:=sub<GL(3,GF(17))| [1,0,0,0,16,0,0,0,16],[13,0,0,0,13,0,0,0,8],[15,0,0,0,2,0,0,0,13] >;

C2×C82 in GAP, Magma, Sage, TeX

C_2\times C_8^2
% in TeX

G:=Group("C2xC8^2");
// GroupNames label

G:=SmallGroup(128,179);
// by ID

G=gap.SmallGroup(128,179);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,136,172]);
// Polycyclic

G:=Group<a,b,c|a^2=b^8=c^8=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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