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G = C8×D17order 272 = 24·17

Direct product of C8 and D17

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8×D17, C1363C2, D34.5C4, C4.12D34, C68.12C22, Dic17.5C4, C174(C2×C8), C173C86C2, C34.8(C2×C4), C2.1(C4×D17), (C4×D17).7C2, SmallGroup(272,4)

Series: Derived Chief Lower central Upper central

C1C17 — C8×D17
C1C17C34C68C4×D17 — C8×D17
C17 — C8×D17
C1C8

Generators and relations for C8×D17
 G = < a,b,c | a8=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C2
17C22
17C4
17C2×C4
17C8
17C2×C8

Smallest permutation representation of C8×D17
On 136 points
Generators in S136
(1 129 64 95 22 107 46 74)(2 130 65 96 23 108 47 75)(3 131 66 97 24 109 48 76)(4 132 67 98 25 110 49 77)(5 133 68 99 26 111 50 78)(6 134 52 100 27 112 51 79)(7 135 53 101 28 113 35 80)(8 136 54 102 29 114 36 81)(9 120 55 86 30 115 37 82)(10 121 56 87 31 116 38 83)(11 122 57 88 32 117 39 84)(12 123 58 89 33 118 40 85)(13 124 59 90 34 119 41 69)(14 125 60 91 18 103 42 70)(15 126 61 92 19 104 43 71)(16 127 62 93 20 105 44 72)(17 128 63 94 21 106 45 73)
(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)(120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(1 21)(2 20)(3 19)(4 18)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(69 99)(70 98)(71 97)(72 96)(73 95)(74 94)(75 93)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 102)(84 101)(85 100)(103 132)(104 131)(105 130)(106 129)(107 128)(108 127)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 136)(117 135)(118 134)(119 133)

G:=sub<Sym(136)| (1,129,64,95,22,107,46,74)(2,130,65,96,23,108,47,75)(3,131,66,97,24,109,48,76)(4,132,67,98,25,110,49,77)(5,133,68,99,26,111,50,78)(6,134,52,100,27,112,51,79)(7,135,53,101,28,113,35,80)(8,136,54,102,29,114,36,81)(9,120,55,86,30,115,37,82)(10,121,56,87,31,116,38,83)(11,122,57,88,32,117,39,84)(12,123,58,89,33,118,40,85)(13,124,59,90,34,119,41,69)(14,125,60,91,18,103,42,70)(15,126,61,92,19,104,43,71)(16,127,62,93,20,105,44,72)(17,128,63,94,21,106,45,73), (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)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,21)(2,20)(3,19)(4,18)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,102)(84,101)(85,100)(103,132)(104,131)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,136)(117,135)(118,134)(119,133)>;

G:=Group( (1,129,64,95,22,107,46,74)(2,130,65,96,23,108,47,75)(3,131,66,97,24,109,48,76)(4,132,67,98,25,110,49,77)(5,133,68,99,26,111,50,78)(6,134,52,100,27,112,51,79)(7,135,53,101,28,113,35,80)(8,136,54,102,29,114,36,81)(9,120,55,86,30,115,37,82)(10,121,56,87,31,116,38,83)(11,122,57,88,32,117,39,84)(12,123,58,89,33,118,40,85)(13,124,59,90,34,119,41,69)(14,125,60,91,18,103,42,70)(15,126,61,92,19,104,43,71)(16,127,62,93,20,105,44,72)(17,128,63,94,21,106,45,73), (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)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,21)(2,20)(3,19)(4,18)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,102)(84,101)(85,100)(103,132)(104,131)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,136)(117,135)(118,134)(119,133) );

G=PermutationGroup([(1,129,64,95,22,107,46,74),(2,130,65,96,23,108,47,75),(3,131,66,97,24,109,48,76),(4,132,67,98,25,110,49,77),(5,133,68,99,26,111,50,78),(6,134,52,100,27,112,51,79),(7,135,53,101,28,113,35,80),(8,136,54,102,29,114,36,81),(9,120,55,86,30,115,37,82),(10,121,56,87,31,116,38,83),(11,122,57,88,32,117,39,84),(12,123,58,89,33,118,40,85),(13,124,59,90,34,119,41,69),(14,125,60,91,18,103,42,70),(15,126,61,92,19,104,43,71),(16,127,62,93,20,105,44,72),(17,128,63,94,21,106,45,73)], [(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),(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,21),(2,20),(3,19),(4,18),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,60),(50,59),(51,58),(69,99),(70,98),(71,97),(72,96),(73,95),(74,94),(75,93),(76,92),(77,91),(78,90),(79,89),(80,88),(81,87),(82,86),(83,102),(84,101),(85,100),(103,132),(104,131),(105,130),(106,129),(107,128),(108,127),(109,126),(110,125),(111,124),(112,123),(113,122),(114,121),(115,120),(116,136),(117,135),(118,134),(119,133)])

80 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H17A···17H34A···34H68A···68P136A···136AF
order122244448888888817···1734···3468···68136···136
size1117171117171111171717172···22···22···22···2

80 irreducible representations

dim11111112222
type++++++
imageC1C2C2C2C4C4C8D17D34C4×D17C8×D17
kernelC8×D17C173C8C136C4×D17Dic17D34D17C8C4C2C1
# reps1111228881632

Matrix representation of C8×D17 in GL3(𝔽137) generated by

9600
01000
00100
,
100
02211
013612
,
100
06677
08471
G:=sub<GL(3,GF(137))| [96,0,0,0,100,0,0,0,100],[1,0,0,0,22,136,0,11,12],[1,0,0,0,66,84,0,77,71] >;

C8×D17 in GAP, Magma, Sage, TeX

C_8\times D_{17}
% in TeX

G:=Group("C8xD17");
// GroupNames label

G:=SmallGroup(272,4);
// by ID

G=gap.SmallGroup(272,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,26,42,6404]);
// Polycyclic

G:=Group<a,b,c|a^8=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8×D17 in TeX

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