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## G = C22×Dic9order 144 = 24·32

### Direct product of C22 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C22×Dic9
 Chief series C1 — C3 — C9 — C18 — Dic9 — C2×Dic9 — C22×Dic9
 Lower central C9 — C22×Dic9
 Upper central C1 — C23

Generators and relations for C22×Dic9
G = < a,b,c,d | a2=b2=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 191 in 81 conjugacy classes, 59 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, C23, C9, Dic3, C2×C6, C22×C4, C18, C18, C2×Dic3, C22×C6, Dic9, C2×C18, C22×Dic3, C2×Dic9, C22×C18, C22×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, D9, C2×Dic3, C22×S3, Dic9, D18, C22×Dic3, C2×Dic9, C22×D9, C22×Dic9

Smallest permutation representation of C22×Dic9
Regular action on 144 points
Generators in S144
(1 54)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 55)(34 56)(35 57)(36 58)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 19)(15 20)(16 21)(17 22)(18 23)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(73 104)(74 105)(75 106)(76 107)(77 108)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 127)(117 128)(118 129)(119 130)(120 131)(121 132)(122 133)(123 134)(124 135)(125 136)(126 137)
(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)
(1 94 10 103)(2 93 11 102)(3 92 12 101)(4 91 13 100)(5 108 14 99)(6 107 15 98)(7 106 16 97)(8 105 17 96)(9 104 18 95)(19 86 28 77)(20 85 29 76)(21 84 30 75)(22 83 31 74)(23 82 32 73)(24 81 33 90)(25 80 34 89)(26 79 35 88)(27 78 36 87)(37 139 46 130)(38 138 47 129)(39 137 48 128)(40 136 49 127)(41 135 50 144)(42 134 51 143)(43 133 52 142)(44 132 53 141)(45 131 54 140)(55 120 64 111)(56 119 65 110)(57 118 66 109)(58 117 67 126)(59 116 68 125)(60 115 69 124)(61 114 70 123)(62 113 71 122)(63 112 72 121)

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

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

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

C22×Dic9 is a maximal subgroup of
C18.C42  C23.16D18  C222Dic18  Dic94D4  C22.4D36  C23.23D18  Dic9⋊D4  C22×C4×D9  Dic9⋊A4
C22×Dic9 is a maximal quotient of
C23.26D18  D4.Dic9

48 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 6A ··· 6G 9A 9B 9C 18A ··· 18U order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 9 9 9 18 ··· 18 size 1 1 ··· 1 2 9 ··· 9 2 ··· 2 2 2 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + - + + - + image C1 C2 C2 C4 S3 Dic3 D6 D9 Dic9 D18 kernel C22×Dic9 C2×Dic9 C22×C18 C2×C18 C22×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 8 1 4 3 3 12 9

Matrix representation of C22×Dic9 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 36 0 0 0 0 6 17 0 0 20 26
,
 36 0 0 0 0 31 0 0 0 0 7 14 0 0 7 30
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,36,0,0,0,0,6,20,0,0,17,26],[36,0,0,0,0,31,0,0,0,0,7,7,0,0,14,30] >;

C22×Dic9 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_9
% in TeX

G:=Group("C2^2xDic9");
// GroupNames label

G:=SmallGroup(144,45);
// by ID

G=gap.SmallGroup(144,45);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^18=1,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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