direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D36⋊5C2, C18.4C24, D36⋊12C22, C36.43C23, D18.1C23, C23.32D18, Dic9.2C23, Dic18⋊11C22, (C2×C4)⋊10D18, (C22×C4)⋊8D9, (C2×D36)⋊14C2, C18⋊1(C4○D4), (C22×C36)⋊8C2, (C4×D9)⋊7C22, C9⋊D4⋊6C22, C2.5(C23×D9), (C2×C36)⋊13C22, (C2×C12).381D6, C6.41(S3×C23), C4.43(C22×D9), (C2×Dic18)⋊15C2, C6.90(C4○D12), (C2×C18).65C23, (C22×C12).28S3, (C22×C6).149D6, C22.5(C22×D9), C12.205(C22×S3), (C22×C18).46C22, (C2×Dic9).45C22, (C22×D9).28C22, C9⋊1(C2×C4○D4), (C2×C4×D9)⋊15C2, C3.(C2×C4○D12), (C2×C9⋊D4)⋊12C2, (C2×C6).222(C22×S3), SmallGroup(288,355)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D36⋊5C2
G = < a,b,c,d | a2=b36=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b18c >
Subgroups: 952 in 246 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, D9, C18, C18, C18, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4○D4, Dic9, C36, D18, D18, C2×C18, C2×C18, C2×C18, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, C2×C36, C22×D9, C22×C18, C2×C4○D12, C2×Dic18, C2×C4×D9, C2×D36, D36⋊5C2, C2×C9⋊D4, C22×C36, C2×D36⋊5C2
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, D9, C22×S3, C2×C4○D4, D18, C4○D12, S3×C23, C22×D9, C2×C4○D12, D36⋊5C2, C23×D9, C2×D36⋊5C2
►On 144 points
Generators in S144
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 144)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)
(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)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 112)(8 111)(9 110)(10 109)(11 144)(12 143)(13 142)(14 141)(15 140)(16 139)(17 138)(18 137)(19 136)(20 135)(21 134)(22 133)(23 132)(24 131)(25 130)(26 129)(27 128)(28 127)(29 126)(30 125)(31 124)(32 123)(33 122)(34 121)(35 120)(36 119)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)
(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 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 141)(80 142)(81 143)(82 144)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)
G:=sub<Sym(144)| (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108), (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), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,112)(8,111)(9,110)(10,109)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95), (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)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)>;
G:=Group( (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108), (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), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,112)(8,111)(9,110)(10,109)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95), (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)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134) );
G=PermutationGroup([[(1,119),(2,120),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,139),(22,140),(23,141),(24,142),(25,143),(26,144),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108)], [(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)], [(1,118),(2,117),(3,116),(4,115),(5,114),(6,113),(7,112),(8,111),(9,110),(10,109),(11,144),(12,143),(13,142),(14,141),(15,140),(16,139),(17,138),(18,137),(19,136),(20,135),(21,134),(22,133),(23,132),(24,131),(25,130),(26,129),(27,128),(28,127),(29,126),(30,125),(31,124),(32,123),(33,122),(34,121),(35,120),(36,119),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,108),(60,107),(61,106),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95)], [(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),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(73,135),(74,136),(75,137),(76,138),(77,139),(78,140),(79,141),(80,142),(81,143),(82,144),(83,109),(84,110),(85,111),(86,112),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,121),(96,122),(97,123),(98,124),(99,125),(100,126),(101,127),(102,128),(103,129),(104,130),(105,131),(106,132),(107,133),(108,134)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6G | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | D9 | D18 | D18 | C4○D12 | D36⋊5C2 |
| kernel | C2×D36⋊5C2 | C2×Dic18 | C2×C4×D9 | C2×D36 | D36⋊5C2 | C2×C9⋊D4 | C22×C36 | C22×C12 | C2×C12 | C22×C6 | C18 | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 6 | 1 | 4 | 3 | 18 | 3 | 8 | 24 |
Matrix representation of C2×D36⋊5C2 ►in GL3(𝔽37) generated by
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 36 | 0 | 0 |
| 0 | 25 | 33 |
| 0 | 4 | 29 |
| 36 | 0 | 0 |
| 0 | 12 | 4 |
| 0 | 29 | 25 |
| 36 | 0 | 0 |
| 0 | 30 | 23 |
| 0 | 14 | 7 |
G:=sub<GL(3,GF(37))| [36,0,0,0,36,0,0,0,36],[36,0,0,0,25,4,0,33,29],[36,0,0,0,12,29,0,4,25],[36,0,0,0,30,14,0,23,7] >;
C2×D36⋊5C2 in GAP, Magma, Sage, TeX
C_2\times D_{36}\rtimes_5C_2 % in TeX
G:=Group("C2xD36:5C2"); // GroupNames label
G:=SmallGroup(288,355);
// by ID
G=gap.SmallGroup(288,355);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^36=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^18*c>;
// generators/relations