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## G = D68⋊C2order 272 = 24·17

### 4th semidirect product of D68 and C2 acting faithfully

Aliases: D684C2, Q82D17, C4.7D34, C34.8C23, C68.7C22, D34.3C22, Dic17.9C22, (C4×D17)⋊3C2, C173(C4○D4), (Q8×C17)⋊3C2, C2.9(C22×D17), SmallGroup(272,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — D68⋊C2
 Chief series C1 — C17 — C34 — D34 — C4×D17 — D68⋊C2
 Lower central C17 — C34 — D68⋊C2
 Upper central C1 — C2 — Q8

Generators and relations for D68⋊C2
G = < a,b,c | a68=b2=c2=1, bab=a-1, cac=a33, cbc=a66b >

Smallest permutation representation of D68⋊C2
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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)(95 136)(96 135)(97 134)(98 133)(99 132)(100 131)(101 130)(102 129)(103 128)(104 127)(105 126)(106 125)(107 124)(108 123)(109 122)(110 121)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 99)(2 132)(3 97)(4 130)(5 95)(6 128)(7 93)(8 126)(9 91)(10 124)(11 89)(12 122)(13 87)(14 120)(15 85)(16 118)(17 83)(18 116)(19 81)(20 114)(21 79)(22 112)(23 77)(24 110)(25 75)(26 108)(27 73)(28 106)(29 71)(30 104)(31 69)(32 102)(33 135)(34 100)(35 133)(36 98)(37 131)(38 96)(39 129)(40 94)(41 127)(42 92)(43 125)(44 90)(45 123)(46 88)(47 121)(48 86)(49 119)(50 84)(51 117)(52 82)(53 115)(54 80)(55 113)(56 78)(57 111)(58 76)(59 109)(60 74)(61 107)(62 72)(63 105)(64 70)(65 103)(66 136)(67 101)(68 134)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 17A ··· 17H 34A ··· 34H 68A ··· 68X order 1 2 2 2 2 4 4 4 4 4 17 ··· 17 34 ··· 34 68 ··· 68 size 1 1 34 34 34 2 2 2 17 17 2 ··· 2 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 2 2 2 4 type + + + + + + + image C1 C2 C2 C2 C4○D4 D17 D34 D68⋊C2 kernel D68⋊C2 C4×D17 D68 Q8×C17 C17 Q8 C4 C1 # reps 1 3 3 1 2 8 24 8

Matrix representation of D68⋊C2 in GL4(𝔽137) generated by

 106 21 0 0 116 107 0 0 0 0 0 1 0 0 136 0
,
 106 21 0 0 13 31 0 0 0 0 0 1 0 0 1 0
,
 136 0 0 0 13 1 0 0 0 0 0 100 0 0 37 0
G:=sub<GL(4,GF(137))| [106,116,0,0,21,107,0,0,0,0,0,136,0,0,1,0],[106,13,0,0,21,31,0,0,0,0,0,1,0,0,1,0],[136,13,0,0,0,1,0,0,0,0,0,37,0,0,100,0] >;

D68⋊C2 in GAP, Magma, Sage, TeX

D_{68}\rtimes C_2
% in TeX

G:=Group("D68:C2");
// GroupNames label

G:=SmallGroup(272,43);
// by ID

G=gap.SmallGroup(272,43);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,46,182,97,42,6404]);
// Polycyclic

G:=Group<a,b,c|a^68=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^33,c*b*c=a^66*b>;
// generators/relations

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