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G = C8⋊D17order 272 = 24·17

3rd semidirect product of C8 and D17 acting via D17/C17=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D17, C1364C2, D34.2C4, C4.13D34, C173M4(2), C68.13C22, Dic17.2C4, C173C84C2, C34.9(C2×C4), C2.3(C4×D17), (C4×D17).2C2, SmallGroup(272,5)

Series: Derived Chief Lower central Upper central

C1C34 — C8⋊D17
C1C17C34C68C4×D17 — C8⋊D17
C17C34 — C8⋊D17
C1C4C8

Generators and relations for C8⋊D17
 G = < a,b,c | a8=b17=c2=1, ab=ba, cac=a5, cbc=b-1 >

34C2
17C22
17C4
2D17
17C2×C4
17C8
17M4(2)

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

74 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D17A···17H34A···34H68A···68P136A···136AF
order122444888817···1734···3468···68136···136
size113411342234342···22···22···22···2

74 irreducible representations

dim11111122222
type++++++
imageC1C2C2C2C4C4M4(2)D17D34C4×D17C8⋊D17
kernelC8⋊D17C173C8C136C4×D17Dic17D34C17C8C4C2C1
# reps1111222881632

Matrix representation of C8⋊D17 in GL2(𝔽137) generated by

5786
5180
,
301
1360
,
01
10
G:=sub<GL(2,GF(137))| [57,51,86,80],[30,136,1,0],[0,1,1,0] >;

C8⋊D17 in GAP, Magma, Sage, TeX

C_8\rtimes D_{17}
% in TeX

G:=Group("C8:D17");
// GroupNames label

G:=SmallGroup(272,5);
// by ID

G=gap.SmallGroup(272,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,101,26,42,6404]);
// Polycyclic

G:=Group<a,b,c|a^8=b^17=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D17 in TeX

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