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## G = C4○D4×C17order 272 = 24·17

### Direct product of C17 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C4○D4×C17, D42C34, Q82C34, C34.13C23, C68.21C22, (C2×C4)⋊3C34, (C2×C68)⋊7C2, (D4×C17)⋊5C2, C4.5(C2×C34), (Q8×C17)⋊5C2, C22.(C2×C34), (C2×C34).2C22, C2.3(C22×C34), SmallGroup(272,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C17
 Chief series C1 — C2 — C34 — C2×C34 — D4×C17 — C4○D4×C17
 Lower central C1 — C2 — C4○D4×C17
 Upper central C1 — C68 — C4○D4×C17

Generators and relations for C4○D4×C17
G = < a,b,c,d | a17=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Smallest permutation representation of C4○D4×C17
On 136 points
Generators in S136
(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 58 33 135)(2 59 34 136)(3 60 18 120)(4 61 19 121)(5 62 20 122)(6 63 21 123)(7 64 22 124)(8 65 23 125)(9 66 24 126)(10 67 25 127)(11 68 26 128)(12 52 27 129)(13 53 28 130)(14 54 29 131)(15 55 30 132)(16 56 31 133)(17 57 32 134)(35 111 102 77)(36 112 86 78)(37 113 87 79)(38 114 88 80)(39 115 89 81)(40 116 90 82)(41 117 91 83)(42 118 92 84)(43 119 93 85)(44 103 94 69)(45 104 95 70)(46 105 96 71)(47 106 97 72)(48 107 98 73)(49 108 99 74)(50 109 100 75)(51 110 101 76)
(1 135 33 58)(2 136 34 59)(3 120 18 60)(4 121 19 61)(5 122 20 62)(6 123 21 63)(7 124 22 64)(8 125 23 65)(9 126 24 66)(10 127 25 67)(11 128 26 68)(12 129 27 52)(13 130 28 53)(14 131 29 54)(15 132 30 55)(16 133 31 56)(17 134 32 57)(35 111 102 77)(36 112 86 78)(37 113 87 79)(38 114 88 80)(39 115 89 81)(40 116 90 82)(41 117 91 83)(42 118 92 84)(43 119 93 85)(44 103 94 69)(45 104 95 70)(46 105 96 71)(47 106 97 72)(48 107 98 73)(49 108 99 74)(50 109 100 75)(51 110 101 76)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 69)(34 70)(35 126)(36 127)(37 128)(38 129)(39 130)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 86)(68 87)

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

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

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

170 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 17A ··· 17P 34A ··· 34P 34Q ··· 34BL 68A ··· 68AF 68AG ··· 68CB order 1 2 2 2 2 4 4 4 4 4 17 ··· 17 34 ··· 34 34 ··· 34 68 ··· 68 68 ··· 68 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

170 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C17 C34 C34 C34 C4○D4 C4○D4×C17 kernel C4○D4×C17 C2×C68 D4×C17 Q8×C17 C4○D4 C2×C4 D4 Q8 C17 C1 # reps 1 3 3 1 16 48 48 16 2 32

Matrix representation of C4○D4×C17 in GL2(𝔽137) generated by

 123 0 0 123
,
 100 0 0 100
,
 37 46 0 100
,
 46 123 63 91
G:=sub<GL(2,GF(137))| [123,0,0,123],[100,0,0,100],[37,0,46,100],[46,63,123,91] >;

C4○D4×C17 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{17}
% in TeX

G:=Group("C4oD4xC17");
// GroupNames label

G:=SmallGroup(272,49);
// by ID

G=gap.SmallGroup(272,49);
# by ID

G:=PCGroup([5,-2,-2,-2,-17,-2,1381,522]);
// Polycyclic

G:=Group<a,b,c,d|a^17=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

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