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G = C22×D36order 288 = 25·32

Direct product of C22 and D36

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

Aliases: C22×D36, C362C23, D181C23, C18.3C24, C23.40D18, (C2×C18)⋊6D4, (C2×C4)⋊9D18, C181(C2×D4), C91(C22×D4), (C22×C4)⋊7D9, C42(C22×D9), C3.(C22×D12), (C22×C36)⋊7C2, C6.47(C2×D12), (C2×C6).37D12, (C23×D9)⋊3C2, C2.4(C23×D9), (C2×C36)⋊12C22, (C2×C12).380D6, C6.40(S3×C23), (C22×C12).22S3, (C2×C18).64C23, (C22×C6).148D6, (C22×D9)⋊5C22, C12.186(C22×S3), C22.30(C22×D9), (C22×C18).45C22, (C2×C6).221(C22×S3), SmallGroup(288,354)

Series: Derived Chief Lower central Upper central

C1C18 — C22×D36
C1C3C9C18D18C22×D9C23×D9 — C22×D36
C9C18 — C22×D36
C1C23C22×C4

Generators and relations for C22×D36
 G = < a,b,c,d | a2=b2=c36=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1784 in 354 conjugacy classes, 132 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], S3 [×8], C6, C6 [×6], C2×C4 [×6], D4 [×16], C23, C23 [×20], C9, C12 [×4], D6 [×32], C2×C6 [×7], C22×C4, C2×D4 [×12], C24 [×2], D9 [×8], C18, C18 [×6], D12 [×16], C2×C12 [×6], C22×S3 [×20], C22×C6, C22×D4, C36 [×4], D18 [×8], D18 [×24], C2×C18 [×7], C2×D12 [×12], C22×C12, S3×C23 [×2], D36 [×16], C2×C36 [×6], C22×D9 [×12], C22×D9 [×8], C22×C18, C22×D12, C2×D36 [×12], C22×C36, C23×D9 [×2], C22×D36
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D9, D12 [×4], C22×S3 [×7], C22×D4, D18 [×7], C2×D12 [×6], S3×C23, D36 [×4], C22×D9 [×7], C22×D12, C2×D36 [×6], C23×D9, C22×D36

Smallest permutation representation of C22×D36
On 144 points
Generators in S144
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 121)(80 122)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 109)(104 110)(105 111)(106 112)(107 113)(108 114)
(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 9)(2 8)(3 7)(4 6)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(37 67)(38 66)(39 65)(40 64)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(68 72)(69 71)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(106 108)(109 117)(110 116)(111 115)(112 114)(118 144)(119 143)(120 142)(121 141)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)

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

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

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

84 conjugacy classes

class 1 2A···2G2H···2O 3 4A4B4C4D6A···6G9A9B9C12A···12H18A···18U36A···36X
order12···22···2344446···699912···1218···1836···36
size11···118···18222222···22222···22···22···2

84 irreducible representations

dim1111222222222
type+++++++++++++
imageC1C2C2C2S3D4D6D6D9D12D18D18D36
kernelC22×D36C2×D36C22×C36C23×D9C22×C12C2×C18C2×C12C22×C6C22×C4C2×C6C2×C4C23C22
# reps1121214613818324

Matrix representation of C22×D36 in GL6(𝔽37)

3600000
010000
0036000
0003600
000010
000001
,
3600000
0360000
0036000
0003600
000010
000001
,
100000
010000
0032500
00322700
00002631
0000620
,
3600000
0360000
00363600
000100
0000620
00002631

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,32,0,0,0,0,5,27,0,0,0,0,0,0,26,6,0,0,0,0,31,20],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,36,1,0,0,0,0,0,0,6,26,0,0,0,0,20,31] >;

C22×D36 in GAP, Magma, Sage, TeX

C_2^2\times D_{36}
% in TeX

G:=Group("C2^2xD36");
// GroupNames label

G:=SmallGroup(288,354);
// by ID

G=gap.SmallGroup(288,354);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^36=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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