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G = C2×C73⋊C3order 438 = 2·3·73

Direct product of C2 and C73⋊C3

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

Aliases: C2×C73⋊C3, C146⋊C3, C732C6, SmallGroup(438,2)

Series: Derived Chief Lower central Upper central

C1C73 — C2×C73⋊C3
C1C73C73⋊C3 — C2×C73⋊C3
C73 — C2×C73⋊C3
C1C2

Generators and relations for C2×C73⋊C3
 G = < a,b,c | a2=b73=c3=1, ab=ba, ac=ca, cbc-1=b64 >

73C3
73C6

Smallest permutation representation of C2×C73⋊C3
On 146 points
Generators in S146
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 145)(73 146)
(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)
(2 9 65)(3 17 56)(4 25 47)(5 33 38)(6 41 29)(7 49 20)(8 57 11)(10 73 66)(12 16 48)(13 24 39)(14 32 30)(15 40 21)(18 64 67)(19 72 58)(22 23 31)(26 55 68)(27 63 59)(28 71 50)(34 46 69)(35 54 60)(36 62 51)(37 70 42)(43 45 61)(44 53 52)(75 82 138)(76 90 129)(77 98 120)(78 106 111)(79 114 102)(80 122 93)(81 130 84)(83 146 139)(85 89 121)(86 97 112)(87 105 103)(88 113 94)(91 137 140)(92 145 131)(95 96 104)(99 128 141)(100 136 132)(101 144 123)(107 119 142)(108 127 133)(109 135 124)(110 143 115)(116 118 134)(117 126 125)

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

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

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

54 conjugacy classes

class 1  2 3A3B6A6B73A···73X146A···146X
order12336673···73146···146
size11737373733···33···3

54 irreducible representations

dim111133
type++
imageC1C2C3C6C73⋊C3C2×C73⋊C3
kernelC2×C73⋊C3C73⋊C3C146C73C2C1
# reps11222424

Matrix representation of C2×C73⋊C3 in GL4(𝔽439) generated by

438000
0100
0010
0001
,
1000
0233131
0100
0010
,
171000
0100
015040055
039642838
G:=sub<GL(4,GF(439))| [438,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,233,1,0,0,13,0,1,0,1,0,0],[171,0,0,0,0,1,150,396,0,0,400,428,0,0,55,38] >;

C2×C73⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{73}\rtimes C_3
% in TeX

G:=Group("C2xC73:C3");
// GroupNames label

G:=SmallGroup(438,2);
// by ID

G=gap.SmallGroup(438,2);
# by ID

G:=PCGroup([3,-2,-3,-73,221]);
// Polycyclic

G:=Group<a,b,c|a^2=b^73=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^64>;
// generators/relations

Export

Subgroup lattice of C2×C73⋊C3 in TeX

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