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G = C2×D68order 272 = 24·17

Direct product of C2 and D68

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D68, C42D34, C341D4, C682C22, D341C22, C34.3C23, C22.10D34, C171(C2×D4), (C2×C68)⋊3C2, (C2×C4)⋊2D17, (C22×D17)⋊1C2, C2.4(C22×D17), (C2×C34).10C22, SmallGroup(272,38)

Series: Derived Chief Lower central Upper central

C1C34 — C2×D68
C1C17C34D34C22×D17 — C2×D68
C17C34 — C2×D68
C1C22C2×C4

Generators and relations for C2×D68
 G = < a,b,c | a2=b68=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 502 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C2×D4, C17, D17, C34, C34, C68, D34, D34, C2×C34, D68, C2×C68, C22×D17, C2×D68
Quotients: C1, C2, C22, D4, C23, C2×D4, D17, D34, D68, C22×D17, C2×D68

Smallest permutation representation of C2×D68
On 136 points
Generators in S136
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 121)(11 122)(12 123)(13 124)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 136)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)
(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 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B17A···17H34A···34X68A···68AF
order122222224417···1734···3468···68
size111134343434222···22···22···2

74 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2D4D17D34D34D68
kernelC2×D68D68C2×C68C22×D17C34C2×C4C4C22C2
# reps14122816832

Matrix representation of C2×D68 in GL3(𝔽137) generated by

13600
01360
00136
,
13600
011694
04388
,
100
011694
010921
G:=sub<GL(3,GF(137))| [136,0,0,0,136,0,0,0,136],[136,0,0,0,116,43,0,94,88],[1,0,0,0,116,109,0,94,21] >;

C2×D68 in GAP, Magma, Sage, TeX

C_2\times D_{68}
% in TeX

G:=Group("C2xD68");
// GroupNames label

G:=SmallGroup(272,38);
// by ID

G=gap.SmallGroup(272,38);
# by ID

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

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

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