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G = C2×C82order 128 = 27

Abelian group of type [2,8,8]

direct product, p-group, abelian, monomial

Aliases: C2×C82, SmallGroup(128,179)

Series: Derived Chief Lower central Upper central Jennings

C1 — C2×C82
C1C2C22C2×C4C42C2×C42C2×C4×C8 — C2×C82
C1 — C2×C82
C1 — C2×C82
C1C22C22C42 — 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···2G4A···4X8A···8CR
order12···24···48···8
size11···11···11···1

128 irreducible representations

dim111111
type+++
imageC1C2C2C4C4C8
kernelC2×C82C82C2×C4×C8C4×C8C22×C8C2×C8
# reps143121296

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

100
0160
0016
,
1300
0130
008
,
1500
020
0013
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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