Copied to
clipboard

G = C2×C79⋊C3order 474 = 2·3·79

Direct product of C2 and C79⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C2×C79⋊C3, C158⋊C3, C792C6, SmallGroup(474,2)

Series: Derived Chief Lower central Upper central

C1C79 — C2×C79⋊C3
C1C79C79⋊C3 — C2×C79⋊C3
C79 — C2×C79⋊C3
C1C2

Generators and relations for C2×C79⋊C3
 G = < a,b,c | a2=b79=c3=1, ab=ba, ac=ca, cbc-1=b55 >

79C3
79C6

Smallest permutation representation of C2×C79⋊C3
On 158 points
Generators in S158
(1 80)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 144)(66 145)(67 146)(68 147)(69 148)(70 149)(71 150)(72 151)(73 152)(74 153)(75 154)(76 155)(77 156)(78 157)(79 158)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158)
(2 24 56)(3 47 32)(4 70 8)(5 14 63)(6 37 39)(7 60 15)(9 27 46)(10 50 22)(11 73 77)(12 17 53)(13 40 29)(16 30 36)(18 76 67)(19 20 43)(21 66 74)(23 33 26)(25 79 57)(28 69 64)(31 59 71)(34 49 78)(35 72 54)(38 62 61)(41 52 68)(42 75 44)(45 65 51)(48 55 58)(81 103 135)(82 126 111)(83 149 87)(84 93 142)(85 116 118)(86 139 94)(88 106 125)(89 129 101)(90 152 156)(91 96 132)(92 119 108)(95 109 115)(97 155 146)(98 99 122)(100 145 153)(102 112 105)(104 158 136)(107 148 143)(110 138 150)(113 128 157)(114 151 133)(117 141 140)(120 131 147)(121 154 123)(124 144 130)(127 134 137)

G:=sub<Sym(158)| (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,144)(66,145)(67,146)(68,147)(69,148)(70,149)(71,150)(72,151)(73,152)(74,153)(75,154)(76,155)(77,156)(78,157)(79,158), (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)(80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158), (2,24,56)(3,47,32)(4,70,8)(5,14,63)(6,37,39)(7,60,15)(9,27,46)(10,50,22)(11,73,77)(12,17,53)(13,40,29)(16,30,36)(18,76,67)(19,20,43)(21,66,74)(23,33,26)(25,79,57)(28,69,64)(31,59,71)(34,49,78)(35,72,54)(38,62,61)(41,52,68)(42,75,44)(45,65,51)(48,55,58)(81,103,135)(82,126,111)(83,149,87)(84,93,142)(85,116,118)(86,139,94)(88,106,125)(89,129,101)(90,152,156)(91,96,132)(92,119,108)(95,109,115)(97,155,146)(98,99,122)(100,145,153)(102,112,105)(104,158,136)(107,148,143)(110,138,150)(113,128,157)(114,151,133)(117,141,140)(120,131,147)(121,154,123)(124,144,130)(127,134,137)>;

G:=Group( (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,144)(66,145)(67,146)(68,147)(69,148)(70,149)(71,150)(72,151)(73,152)(74,153)(75,154)(76,155)(77,156)(78,157)(79,158), (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)(80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158), (2,24,56)(3,47,32)(4,70,8)(5,14,63)(6,37,39)(7,60,15)(9,27,46)(10,50,22)(11,73,77)(12,17,53)(13,40,29)(16,30,36)(18,76,67)(19,20,43)(21,66,74)(23,33,26)(25,79,57)(28,69,64)(31,59,71)(34,49,78)(35,72,54)(38,62,61)(41,52,68)(42,75,44)(45,65,51)(48,55,58)(81,103,135)(82,126,111)(83,149,87)(84,93,142)(85,116,118)(86,139,94)(88,106,125)(89,129,101)(90,152,156)(91,96,132)(92,119,108)(95,109,115)(97,155,146)(98,99,122)(100,145,153)(102,112,105)(104,158,136)(107,148,143)(110,138,150)(113,128,157)(114,151,133)(117,141,140)(120,131,147)(121,154,123)(124,144,130)(127,134,137) );

G=PermutationGroup([(1,80),(2,81),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,92),(14,93),(15,94),(16,95),(17,96),(18,97),(19,98),(20,99),(21,100),(22,101),(23,102),(24,103),(25,104),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,115),(37,116),(38,117),(39,118),(40,119),(41,120),(42,121),(43,122),(44,123),(45,124),(46,125),(47,126),(48,127),(49,128),(50,129),(51,130),(52,131),(53,132),(54,133),(55,134),(56,135),(57,136),(58,137),(59,138),(60,139),(61,140),(62,141),(63,142),(64,143),(65,144),(66,145),(67,146),(68,147),(69,148),(70,149),(71,150),(72,151),(73,152),(74,153),(75,154),(76,155),(77,156),(78,157),(79,158)], [(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),(80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158)], [(2,24,56),(3,47,32),(4,70,8),(5,14,63),(6,37,39),(7,60,15),(9,27,46),(10,50,22),(11,73,77),(12,17,53),(13,40,29),(16,30,36),(18,76,67),(19,20,43),(21,66,74),(23,33,26),(25,79,57),(28,69,64),(31,59,71),(34,49,78),(35,72,54),(38,62,61),(41,52,68),(42,75,44),(45,65,51),(48,55,58),(81,103,135),(82,126,111),(83,149,87),(84,93,142),(85,116,118),(86,139,94),(88,106,125),(89,129,101),(90,152,156),(91,96,132),(92,119,108),(95,109,115),(97,155,146),(98,99,122),(100,145,153),(102,112,105),(104,158,136),(107,148,143),(110,138,150),(113,128,157),(114,151,133),(117,141,140),(120,131,147),(121,154,123),(124,144,130),(127,134,137)])

58 conjugacy classes

class 1  2 3A3B6A6B79A···79Z158A···158Z
order12336679···79158···158
size11797979793···33···3

58 irreducible representations

dim111133
type++
imageC1C2C3C6C79⋊C3C2×C79⋊C3
kernelC2×C79⋊C3C79⋊C3C158C79C2C1
# reps11222626

Matrix representation of C2×C79⋊C3 in GL3(𝔽1423) generated by

142200
014220
001422
,
5412511
515441068
3022701271
,
11943741175
356962725
120256690
G:=sub<GL(3,GF(1423))| [1422,0,0,0,1422,0,0,0,1422],[541,51,302,251,544,270,1,1068,1271],[1194,356,1202,374,962,56,1175,725,690] >;

C2×C79⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{79}\rtimes C_3
% in TeX

G:=Group("C2xC79:C3");
// GroupNames label

G:=SmallGroup(474,2);
// by ID

G=gap.SmallGroup(474,2);
# by ID

G:=PCGroup([3,-2,-3,-79,626]);
// Polycyclic

G:=Group<a,b,c|a^2=b^79=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^55>;
// generators/relations

Export

Subgroup lattice of C2×C79⋊C3 in TeX

׿
×
𝔽