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G = C152⋊C2order 304 = 24·19

2nd semidirect product of C152 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82D19, C1522C2, C38.1D4, C4.8D38, C2.3D76, C191SD16, D76.1C2, Dic381C2, C76.8C22, SmallGroup(304,5)

Series: Derived Chief Lower central Upper central

C1C76 — C152⋊C2
C1C19C38C76D76 — C152⋊C2
C19C38C76 — C152⋊C2
C1C2C4C8

Generators and relations for C152⋊C2
 G = < a,b | a152=b2=1, bab=a75 >

76C2
38C22
38C4
4D19
19Q8
19D4
2Dic19
2D38
19SD16

Smallest permutation representation of C152⋊C2
On 152 points
Generators in S152
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(2 76)(3 151)(4 74)(5 149)(6 72)(7 147)(8 70)(9 145)(10 68)(11 143)(12 66)(13 141)(14 64)(15 139)(16 62)(17 137)(18 60)(19 135)(20 58)(21 133)(22 56)(23 131)(24 54)(25 129)(26 52)(27 127)(28 50)(29 125)(30 48)(31 123)(32 46)(33 121)(34 44)(35 119)(36 42)(37 117)(38 40)(39 115)(41 113)(43 111)(45 109)(47 107)(49 105)(51 103)(53 101)(55 99)(57 97)(59 95)(61 93)(63 91)(65 89)(67 87)(69 85)(71 83)(73 81)(75 79)(78 152)(80 150)(82 148)(84 146)(86 144)(88 142)(90 140)(92 138)(94 136)(96 134)(98 132)(100 130)(102 128)(104 126)(106 124)(108 122)(110 120)(112 118)(114 116)

G:=sub<Sym(152)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (2,76)(3,151)(4,74)(5,149)(6,72)(7,147)(8,70)(9,145)(10,68)(11,143)(12,66)(13,141)(14,64)(15,139)(16,62)(17,137)(18,60)(19,135)(20,58)(21,133)(22,56)(23,131)(24,54)(25,129)(26,52)(27,127)(28,50)(29,125)(30,48)(31,123)(32,46)(33,121)(34,44)(35,119)(36,42)(37,117)(38,40)(39,115)(41,113)(43,111)(45,109)(47,107)(49,105)(51,103)(53,101)(55,99)(57,97)(59,95)(61,93)(63,91)(65,89)(67,87)(69,85)(71,83)(73,81)(75,79)(78,152)(80,150)(82,148)(84,146)(86,144)(88,142)(90,140)(92,138)(94,136)(96,134)(98,132)(100,130)(102,128)(104,126)(106,124)(108,122)(110,120)(112,118)(114,116)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (2,76)(3,151)(4,74)(5,149)(6,72)(7,147)(8,70)(9,145)(10,68)(11,143)(12,66)(13,141)(14,64)(15,139)(16,62)(17,137)(18,60)(19,135)(20,58)(21,133)(22,56)(23,131)(24,54)(25,129)(26,52)(27,127)(28,50)(29,125)(30,48)(31,123)(32,46)(33,121)(34,44)(35,119)(36,42)(37,117)(38,40)(39,115)(41,113)(43,111)(45,109)(47,107)(49,105)(51,103)(53,101)(55,99)(57,97)(59,95)(61,93)(63,91)(65,89)(67,87)(69,85)(71,83)(73,81)(75,79)(78,152)(80,150)(82,148)(84,146)(86,144)(88,142)(90,140)(92,138)(94,136)(96,134)(98,132)(100,130)(102,128)(104,126)(106,124)(108,122)(110,120)(112,118)(114,116) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(2,76),(3,151),(4,74),(5,149),(6,72),(7,147),(8,70),(9,145),(10,68),(11,143),(12,66),(13,141),(14,64),(15,139),(16,62),(17,137),(18,60),(19,135),(20,58),(21,133),(22,56),(23,131),(24,54),(25,129),(26,52),(27,127),(28,50),(29,125),(30,48),(31,123),(32,46),(33,121),(34,44),(35,119),(36,42),(37,117),(38,40),(39,115),(41,113),(43,111),(45,109),(47,107),(49,105),(51,103),(53,101),(55,99),(57,97),(59,95),(61,93),(63,91),(65,89),(67,87),(69,85),(71,83),(73,81),(75,79),(78,152),(80,150),(82,148),(84,146),(86,144),(88,142),(90,140),(92,138),(94,136),(96,134),(98,132),(100,130),(102,128),(104,126),(106,124),(108,122),(110,120),(112,118),(114,116)])

79 conjugacy classes

class 1 2A2B4A4B8A8B19A···19I38A···38I76A···76R152A···152AJ
order122448819···1938···3876···76152···152
size1176276222···22···22···22···2

79 irreducible representations

dim1111222222
type++++++++
imageC1C2C2C2D4SD16D19D38D76C152⋊C2
kernelC152⋊C2C152Dic38D76C38C19C8C4C2C1
# reps111112991836

Matrix representation of C152⋊C2 in GL2(𝔽457) generated by

385101
307166
,
370100
3487
G:=sub<GL(2,GF(457))| [385,307,101,166],[370,34,100,87] >;

C152⋊C2 in GAP, Magma, Sage, TeX

C_{152}\rtimes C_2
% in TeX

G:=Group("C152:C2");
// GroupNames label

G:=SmallGroup(304,5);
// by ID

G=gap.SmallGroup(304,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-19,61,26,182,42,7204]);
// Polycyclic

G:=Group<a,b|a^152=b^2=1,b*a*b=a^75>;
// generators/relations

Export

Subgroup lattice of C152⋊C2 in TeX

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