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G = A4×C37order 444 = 22·3·37

Direct product of C37 and A4

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

Aliases: A4×C37, C22⋊C111, (C2×C74)⋊1C3, SmallGroup(444,13)

Series: Derived Chief Lower central Upper central

C1C22 — A4×C37
C1C22C2×C74 — A4×C37
C22 — A4×C37
C1C37

Generators and relations for A4×C37
 G = < a,b,c,d | a37=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

3C2
4C3
3C74
4C111

Smallest permutation representation of A4×C37
On 148 points
Generators in S148
(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)
(1 144)(2 145)(3 146)(4 147)(5 148)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 141)(36 142)(37 143)(38 107)(39 108)(40 109)(41 110)(42 111)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)
(38 107 125)(39 108 126)(40 109 127)(41 110 128)(42 111 129)(43 75 130)(44 76 131)(45 77 132)(46 78 133)(47 79 134)(48 80 135)(49 81 136)(50 82 137)(51 83 138)(52 84 139)(53 85 140)(54 86 141)(55 87 142)(56 88 143)(57 89 144)(58 90 145)(59 91 146)(60 92 147)(61 93 148)(62 94 112)(63 95 113)(64 96 114)(65 97 115)(66 98 116)(67 99 117)(68 100 118)(69 101 119)(70 102 120)(71 103 121)(72 104 122)(73 105 123)(74 106 124)

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

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

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

148 conjugacy classes

class 1  2 3A3B37A···37AJ74A···74AJ111A···111BT
order123337···3774···74111···111
size13441···13···34···4

148 irreducible representations

dim111133
type++
imageC1C3C37C111A4A4×C37
kernelA4×C37C2×C74A4C22C37C1
# reps123672136

Matrix representation of A4×C37 in GL3(𝔽223) generated by

13600
01360
00136
,
22200
22201
22210
,
01222
10222
00222
,
010
001
100
G:=sub<GL(3,GF(223))| [136,0,0,0,136,0,0,0,136],[222,222,222,0,0,1,0,1,0],[0,1,0,1,0,0,222,222,222],[0,0,1,1,0,0,0,1,0] >;

A4×C37 in GAP, Magma, Sage, TeX

A_4\times C_{37}
% in TeX

G:=Group("A4xC37");
// GroupNames label

G:=SmallGroup(444,13);
// by ID

G=gap.SmallGroup(444,13);
# by ID

G:=PCGroup([4,-3,-37,-2,2,2666,5331]);
// Polycyclic

G:=Group<a,b,c,d|a^37=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×C37 in TeX

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