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G = D685C2order 272 = 24·17

The semidirect product of D68 and C2 acting through Inn(D68)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D685C2, C4.16D34, Dic345C2, C34.4C23, C22.2D34, C68.16C22, D34.1C22, Dic17.2C22, (C2×C68)⋊4C2, (C2×C4)⋊3D17, (C4×D17)⋊4C2, C171(C4○D4), C17⋊D43C2, C2.5(C22×D17), (C2×C34).11C22, SmallGroup(272,39)

Series: Derived Chief Lower central Upper central

C1C34 — D685C2
C1C17C34D34C4×D17 — D685C2
C17C34 — D685C2
C1C4C2×C4

Generators and relations for D685C2
 G = < a,b,c | a68=b2=c2=1, bab=a-1, ac=ca, cbc=a34b >

2C2
34C2
34C2
17C4
17C4
17C22
17C22
2C34
2D17
2D17
17C2×C4
17D4
17D4
17D4
17C2×C4
17Q8
17C4○D4

Smallest permutation representation of D685C2
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 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 130)(91 129)(92 128)(93 127)(94 126)(95 125)(96 124)(97 123)(98 122)(99 121)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)
(1 136)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E17A···17H34A···34X68A···68AF
order122224444417···1734···3468···68
size112343411234342···22···22···2

74 irreducible representations

dim11111122222
type+++++++++
imageC1C2C2C2C2C2C4○D4D17D34D34D685C2
kernelD685C2Dic34C4×D17D68C17⋊D4C2×C68C17C2×C4C4C22C1
# reps1121212816832

Matrix representation of D685C2 in GL2(𝔽137) generated by

4395
7928
,
75116
4662
,
12756
12310
G:=sub<GL(2,GF(137))| [43,79,95,28],[75,46,116,62],[127,123,56,10] >;

D685C2 in GAP, Magma, Sage, TeX

D_{68}\rtimes_5C_2
% in TeX

G:=Group("D68:5C2");
// GroupNames label

G:=SmallGroup(272,39);
// by ID

G=gap.SmallGroup(272,39);
# by ID

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

G:=Group<a,b,c|a^68=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^34*b>;
// generators/relations

Export

Subgroup lattice of D685C2 in TeX

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