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G = C22×Dic9order 144 = 24·32

Direct product of C22 and Dic9

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C22×Dic9, C23.3D9, C18.9C23, C22.11D18, C182(C2×C4), (C2×C18)⋊3C4, C92(C22×C4), (C2×C6).28D6, (C22×C6).7S3, (C2×C6).6Dic3, C2.2(C22×D9), C3.(C22×Dic3), C6.27(C22×S3), (C22×C18).3C2, C6.11(C2×Dic3), (C2×C18).12C22, SmallGroup(144,45)

Series: Derived Chief Lower central Upper central

C1C9 — C22×Dic9
C1C3C9C18Dic9C2×Dic9 — C22×Dic9
C9 — C22×Dic9
C1C23

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 [×6], C3, C4 [×4], C22 [×7], C6, C6 [×6], C2×C4 [×6], C23, C9, Dic3 [×4], C2×C6 [×7], C22×C4, C18, C18 [×6], C2×Dic3 [×6], C22×C6, Dic9 [×4], C2×C18 [×7], C22×Dic3, C2×Dic9 [×6], C22×C18, C22×Dic9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, D9, C2×Dic3 [×6], C22×S3, Dic9 [×4], D18 [×3], C22×Dic3, C2×Dic9 [×6], C22×D9, C22×Dic9

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

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

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

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

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···4H6A···6G9A9B9C18A···18U
order12···234···46···699918···18
size11···129···92···22222···2

48 irreducible representations

dim1111222222
type++++-++-+
imageC1C2C2C4S3Dic3D6D9Dic9D18
kernelC22×Dic9C2×Dic9C22×C18C2×C18C22×C6C2×C6C2×C6C23C22C22
# reps16181433129

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

36000
03600
0010
0001
,
1000
03600
0010
0001
,
1000
03600
00617
002026
,
36000
03100
00714
00730
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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