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G = A4×C41order 492 = 22·3·41

Direct product of C41 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: A4×C41, C22⋊C123, (C2×C82)⋊C3, SmallGroup(492,8)

Series: Derived Chief Lower central Upper central

C1C22 — A4×C41
C1C22C2×C82 — A4×C41
C22 — A4×C41
C1C41

Generators and relations for A4×C41
 G = < a,b,c,d | a41=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

3C2
4C3
3C82
4C123

Smallest permutation representation of A4×C41
On 164 points
Generators in S164
(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)(124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 42)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)(115 164)(116 124)(117 125)(118 126)(119 127)(120 128)(121 129)(122 130)(123 131)
(1 120)(2 121)(3 122)(4 123)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 141)(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 124)(81 125)(82 126)
(42 119 127)(43 120 128)(44 121 129)(45 122 130)(46 123 131)(47 83 132)(48 84 133)(49 85 134)(50 86 135)(51 87 136)(52 88 137)(53 89 138)(54 90 139)(55 91 140)(56 92 141)(57 93 142)(58 94 143)(59 95 144)(60 96 145)(61 97 146)(62 98 147)(63 99 148)(64 100 149)(65 101 150)(66 102 151)(67 103 152)(68 104 153)(69 105 154)(70 106 155)(71 107 156)(72 108 157)(73 109 158)(74 110 159)(75 111 160)(76 112 161)(77 113 162)(78 114 163)(79 115 164)(80 116 124)(81 117 125)(82 118 126)

G:=sub<Sym(164)| (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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,42)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,124)(117,125)(118,126)(119,127)(120,128)(121,129)(122,130)(123,131), (1,120)(2,121)(3,122)(4,123)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,141)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,124)(81,125)(82,126), (42,119,127)(43,120,128)(44,121,129)(45,122,130)(46,123,131)(47,83,132)(48,84,133)(49,85,134)(50,86,135)(51,87,136)(52,88,137)(53,89,138)(54,90,139)(55,91,140)(56,92,141)(57,93,142)(58,94,143)(59,95,144)(60,96,145)(61,97,146)(62,98,147)(63,99,148)(64,100,149)(65,101,150)(66,102,151)(67,103,152)(68,104,153)(69,105,154)(70,106,155)(71,107,156)(72,108,157)(73,109,158)(74,110,159)(75,111,160)(76,112,161)(77,113,162)(78,114,163)(79,115,164)(80,116,124)(81,117,125)(82,118,126)>;

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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,42)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,124)(117,125)(118,126)(119,127)(120,128)(121,129)(122,130)(123,131), (1,120)(2,121)(3,122)(4,123)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,141)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,124)(81,125)(82,126), (42,119,127)(43,120,128)(44,121,129)(45,122,130)(46,123,131)(47,83,132)(48,84,133)(49,85,134)(50,86,135)(51,87,136)(52,88,137)(53,89,138)(54,90,139)(55,91,140)(56,92,141)(57,93,142)(58,94,143)(59,95,144)(60,96,145)(61,97,146)(62,98,147)(63,99,148)(64,100,149)(65,101,150)(66,102,151)(67,103,152)(68,104,153)(69,105,154)(70,106,155)(71,107,156)(72,108,157)(73,109,158)(74,110,159)(75,111,160)(76,112,161)(77,113,162)(78,114,163)(79,115,164)(80,116,124)(81,117,125)(82,118,126) );

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),(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,42),(83,132),(84,133),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149),(101,150),(102,151),(103,152),(104,153),(105,154),(106,155),(107,156),(108,157),(109,158),(110,159),(111,160),(112,161),(113,162),(114,163),(115,164),(116,124),(117,125),(118,126),(119,127),(120,128),(121,129),(122,130),(123,131)], [(1,120),(2,121),(3,122),(4,123),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,119),(42,127),(43,128),(44,129),(45,130),(46,131),(47,132),(48,133),(49,134),(50,135),(51,136),(52,137),(53,138),(54,139),(55,140),(56,141),(57,142),(58,143),(59,144),(60,145),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,164),(80,124),(81,125),(82,126)], [(42,119,127),(43,120,128),(44,121,129),(45,122,130),(46,123,131),(47,83,132),(48,84,133),(49,85,134),(50,86,135),(51,87,136),(52,88,137),(53,89,138),(54,90,139),(55,91,140),(56,92,141),(57,93,142),(58,94,143),(59,95,144),(60,96,145),(61,97,146),(62,98,147),(63,99,148),(64,100,149),(65,101,150),(66,102,151),(67,103,152),(68,104,153),(69,105,154),(70,106,155),(71,107,156),(72,108,157),(73,109,158),(74,110,159),(75,111,160),(76,112,161),(77,113,162),(78,114,163),(79,115,164),(80,116,124),(81,117,125),(82,118,126)])

164 conjugacy classes

class 1  2 3A3B41A···41AN82A···82AN123A···123CB
order123341···4182···82123···123
size13441···13···34···4

164 irreducible representations

dim111133
type++
imageC1C3C41C123A4A4×C41
kernelA4×C41C2×C82A4C22C41C1
# reps124080140

Matrix representation of A4×C41 in GL3(𝔽739) generated by

40000
04000
00400
,
73800
73801
73810
,
01738
10738
00738
,
010
001
100
G:=sub<GL(3,GF(739))| [400,0,0,0,400,0,0,0,400],[738,738,738,0,0,1,0,1,0],[0,1,0,1,0,0,738,738,738],[0,0,1,1,0,0,0,1,0] >;

A4×C41 in GAP, Magma, Sage, TeX

A_4\times C_{41}
% in TeX

G:=Group("A4xC41");
// GroupNames label

G:=SmallGroup(492,8);
// by ID

G=gap.SmallGroup(492,8);
# by ID

G:=PCGroup([4,-3,-41,-2,2,2954,5907]);
// Polycyclic

G:=Group<a,b,c,d|a^41=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

Export

Subgroup lattice of A4×C41 in TeX

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