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G = C136⋊C2order 272 = 24·17

2nd semidirect product of C136 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82D17, C1362C2, C4.8D34, C34.1D4, C2.3D68, C171SD16, D68.1C2, Dic341C2, C68.8C22, SmallGroup(272,6)

Series: Derived Chief Lower central Upper central

C1C68 — C136⋊C2
C1C17C34C68D68 — C136⋊C2
C17C34C68 — C136⋊C2
C1C2C4C8

Generators and relations for C136⋊C2
 G = < a,b | a136=b2=1, bab=a67 >

68C2
34C22
34C4
4D17
17Q8
17D4
2Dic17
2D34
17SD16

Smallest permutation representation of C136⋊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)
(2 68)(3 135)(4 66)(5 133)(6 64)(7 131)(8 62)(9 129)(10 60)(11 127)(12 58)(13 125)(14 56)(15 123)(16 54)(17 121)(18 52)(19 119)(20 50)(21 117)(22 48)(23 115)(24 46)(25 113)(26 44)(27 111)(28 42)(29 109)(30 40)(31 107)(32 38)(33 105)(34 36)(35 103)(37 101)(39 99)(41 97)(43 95)(45 93)(47 91)(49 89)(51 87)(53 85)(55 83)(57 81)(59 79)(61 77)(63 75)(65 73)(67 71)(70 136)(72 134)(74 132)(76 130)(78 128)(80 126)(82 124)(84 122)(86 120)(88 118)(90 116)(92 114)(94 112)(96 110)(98 108)(100 106)(102 104)

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

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

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

71 conjugacy classes

class 1 2A2B4A4B8A8B17A···17H34A···34H68A···68P136A···136AF
order122448817···1734···3468···68136···136
size1168268222···22···22···22···2

71 irreducible representations

dim1111222222
type++++++++
imageC1C2C2C2D4SD16D17D34D68C136⋊C2
kernelC136⋊C2C136Dic34D68C34C17C8C4C2C1
# reps111112881632

Matrix representation of C136⋊C2 in GL2(𝔽137) generated by

2369
6861
,
10
76136
G:=sub<GL(2,GF(137))| [23,68,69,61],[1,76,0,136] >;

C136⋊C2 in GAP, Magma, Sage, TeX

C_{136}\rtimes C_2
% in TeX

G:=Group("C136:C2");
// GroupNames label

G:=SmallGroup(272,6);
// by ID

G=gap.SmallGroup(272,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,61,26,182,42,6404]);
// Polycyclic

G:=Group<a,b|a^136=b^2=1,b*a*b=a^67>;
// generators/relations

Export

Subgroup lattice of C136⋊C2 in TeX

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