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G = C144⋊C2order 288 = 25·32

2nd semidirect product of C144 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C162D9, C1442C2, C91SD32, C48.4S3, C2.4D72, C18.2D8, C4.2D36, C6.2D24, D72.1C2, C36.25D4, C8.14D18, C24.68D6, Dic361C2, C12.35D12, C72.15C22, C3.(C48⋊C2), SmallGroup(288,7)

Series: Derived Chief Lower central Upper central

C1C72 — C144⋊C2
C1C3C9C18C36C72D72 — C144⋊C2
C9C18C36C72 — C144⋊C2
C1C2C4C8C16

Generators and relations for C144⋊C2
 G = < a,b | a144=b2=1, bab=a71 >

72C2
36C22
36C4
24S3
18Q8
18D4
12D6
12Dic3
8D9
9Q16
9D8
6D12
6Dic6
4D18
4Dic9
9SD32
3Dic12
3D24
2Dic18
2D36
3C48⋊C2

Smallest permutation representation of C144⋊C2
On 144 points
Generators in S144
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(2 72)(3 143)(4 70)(5 141)(6 68)(7 139)(8 66)(9 137)(10 64)(11 135)(12 62)(13 133)(14 60)(15 131)(16 58)(17 129)(18 56)(19 127)(20 54)(21 125)(22 52)(23 123)(24 50)(25 121)(26 48)(27 119)(28 46)(29 117)(30 44)(31 115)(32 42)(33 113)(34 40)(35 111)(36 38)(37 109)(39 107)(41 105)(43 103)(45 101)(47 99)(49 97)(51 95)(53 93)(55 91)(57 89)(59 87)(61 85)(63 83)(65 81)(67 79)(69 77)(71 75)(74 144)(76 142)(78 140)(80 138)(82 136)(84 134)(86 132)(88 130)(90 128)(92 126)(94 124)(96 122)(98 120)(100 118)(102 116)(104 114)(106 112)(108 110)

G:=sub<Sym(144)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (2,72)(3,143)(4,70)(5,141)(6,68)(7,139)(8,66)(9,137)(10,64)(11,135)(12,62)(13,133)(14,60)(15,131)(16,58)(17,129)(18,56)(19,127)(20,54)(21,125)(22,52)(23,123)(24,50)(25,121)(26,48)(27,119)(28,46)(29,117)(30,44)(31,115)(32,42)(33,113)(34,40)(35,111)(36,38)(37,109)(39,107)(41,105)(43,103)(45,101)(47,99)(49,97)(51,95)(53,93)(55,91)(57,89)(59,87)(61,85)(63,83)(65,81)(67,79)(69,77)(71,75)(74,144)(76,142)(78,140)(80,138)(82,136)(84,134)(86,132)(88,130)(90,128)(92,126)(94,124)(96,122)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (2,72)(3,143)(4,70)(5,141)(6,68)(7,139)(8,66)(9,137)(10,64)(11,135)(12,62)(13,133)(14,60)(15,131)(16,58)(17,129)(18,56)(19,127)(20,54)(21,125)(22,52)(23,123)(24,50)(25,121)(26,48)(27,119)(28,46)(29,117)(30,44)(31,115)(32,42)(33,113)(34,40)(35,111)(36,38)(37,109)(39,107)(41,105)(43,103)(45,101)(47,99)(49,97)(51,95)(53,93)(55,91)(57,89)(59,87)(61,85)(63,83)(65,81)(67,79)(69,77)(71,75)(74,144)(76,142)(78,140)(80,138)(82,136)(84,134)(86,132)(88,130)(90,128)(92,126)(94,124)(96,122)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(2,72),(3,143),(4,70),(5,141),(6,68),(7,139),(8,66),(9,137),(10,64),(11,135),(12,62),(13,133),(14,60),(15,131),(16,58),(17,129),(18,56),(19,127),(20,54),(21,125),(22,52),(23,123),(24,50),(25,121),(26,48),(27,119),(28,46),(29,117),(30,44),(31,115),(32,42),(33,113),(34,40),(35,111),(36,38),(37,109),(39,107),(41,105),(43,103),(45,101),(47,99),(49,97),(51,95),(53,93),(55,91),(57,89),(59,87),(61,85),(63,83),(65,81),(67,79),(69,77),(71,75),(74,144),(76,142),(78,140),(80,138),(82,136),(84,134),(86,132),(88,130),(90,128),(92,126),(94,124),(96,122),(98,120),(100,118),(102,116),(104,114),(106,112),(108,110)])

75 conjugacy classes

class 1 2A2B 3 4A4B 6 8A8B9A9B9C12A12B16A16B16C16D18A18B18C24A24B24C24D36A···36F48A···48H72A···72L144A···144X
order1223446889991212161616161818182424242436···3648···4872···72144···144
size1172227222222222222222222222···22···22···22···2

75 irreducible representations

dim11112222222222222
type++++++++++++++
imageC1C2C2C2S3D4D6D8D9D12SD32D18D24D36C48⋊C2D72C144⋊C2
kernelC144⋊C2C144Dic36D72C48C36C24C18C16C12C9C8C6C4C3C2C1
# reps1111111232434681224

Matrix representation of C144⋊C2 in GL2(𝔽433) generated by

287244
18943
,
10
432432
G:=sub<GL(2,GF(433))| [287,189,244,43],[1,432,0,432] >;

C144⋊C2 in GAP, Magma, Sage, TeX

C_{144}\rtimes C_2
% in TeX

G:=Group("C144:C2");
// GroupNames label

G:=SmallGroup(288,7);
// by ID

G=gap.SmallGroup(288,7);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,590,58,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b|a^144=b^2=1,b*a*b=a^71>;
// generators/relations

Export

Subgroup lattice of C144⋊C2 in TeX

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