direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D36, C36⋊5D4, C42⋊5D9, C12.55D12, C9⋊1(C4×D4), C4⋊2(C4×D9), C3.(C4×D12), C36⋊4(C2×C4), (C4×C36)⋊7C2, D18⋊1(C2×C4), C2.1(C2×D36), C18.2(C2×D4), D18⋊C4⋊17C2, C12.58(C4×S3), (C4×C12).13S3, C4⋊Dic9⋊16C2, (C2×C4).97D18, C6.31(C2×D12), (C2×D36).10C2, C18.4(C4○D4), (C2×C12).368D6, C18.4(C22×C4), C6.74(C4○D12), (C2×C36).72C22, (C2×C18).14C23, C2.3(D36⋊5C2), C22.11(C22×D9), (C2×Dic9).22C22, (C22×D9).15C22, (C2×C4×D9)⋊7C2, C2.6(C2×C4×D9), C6.43(S3×C2×C4), (C2×C6).171(C22×S3), SmallGroup(288,83)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D36
G = < a,b,c | a4=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 656 in 141 conjugacy classes, 58 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C4×D4, Dic9, C36, C36, D18, D18, C2×C18, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C4×D9, D36, C2×Dic9, C2×C36, C22×D9, C4×D12, C4⋊Dic9, D18⋊C4, C4×C36, C2×C4×D9, C2×D36, C4×D36
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, D9, C4×S3, D12, C22×S3, C4×D4, D18, S3×C2×C4, C2×D12, C4○D12, C4×D9, D36, C22×D9, C4×D12, C2×C4×D9, C2×D36, D36⋊5C2, C4×D36
►On 144 points
Generators in S144
(1 85 127 70)(2 86 128 71)(3 87 129 72)(4 88 130 37)(5 89 131 38)(6 90 132 39)(7 91 133 40)(8 92 134 41)(9 93 135 42)(10 94 136 43)(11 95 137 44)(12 96 138 45)(13 97 139 46)(14 98 140 47)(15 99 141 48)(16 100 142 49)(17 101 143 50)(18 102 144 51)(19 103 109 52)(20 104 110 53)(21 105 111 54)(22 106 112 55)(23 107 113 56)(24 108 114 57)(25 73 115 58)(26 74 116 59)(27 75 117 60)(28 76 118 61)(29 77 119 62)(30 78 120 63)(31 79 121 64)(32 80 122 65)(33 81 123 66)(34 82 124 67)(35 83 125 68)(36 84 126 69)
(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)
(1 117)(2 116)(3 115)(4 114)(5 113)(6 112)(7 111)(8 110)(9 109)(10 144)(11 143)(12 142)(13 141)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(21 133)(22 132)(23 131)(24 130)(25 129)(26 128)(27 127)(28 126)(29 125)(30 124)(31 123)(32 122)(33 121)(34 120)(35 119)(36 118)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)
G:=sub<Sym(144)| (1,85,127,70)(2,86,128,71)(3,87,129,72)(4,88,130,37)(5,89,131,38)(6,90,132,39)(7,91,133,40)(8,92,134,41)(9,93,135,42)(10,94,136,43)(11,95,137,44)(12,96,138,45)(13,97,139,46)(14,98,140,47)(15,99,141,48)(16,100,142,49)(17,101,143,50)(18,102,144,51)(19,103,109,52)(20,104,110,53)(21,105,111,54)(22,106,112,55)(23,107,113,56)(24,108,114,57)(25,73,115,58)(26,74,116,59)(27,75,117,60)(28,76,118,61)(29,77,119,62)(30,78,120,63)(31,79,121,64)(32,80,122,65)(33,81,123,66)(34,82,124,67)(35,83,125,68)(36,84,126,69), (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), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,110)(9,109)(10,144)(11,143)(12,142)(13,141)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,120)(35,119)(36,118)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)>;
G:=Group( (1,85,127,70)(2,86,128,71)(3,87,129,72)(4,88,130,37)(5,89,131,38)(6,90,132,39)(7,91,133,40)(8,92,134,41)(9,93,135,42)(10,94,136,43)(11,95,137,44)(12,96,138,45)(13,97,139,46)(14,98,140,47)(15,99,141,48)(16,100,142,49)(17,101,143,50)(18,102,144,51)(19,103,109,52)(20,104,110,53)(21,105,111,54)(22,106,112,55)(23,107,113,56)(24,108,114,57)(25,73,115,58)(26,74,116,59)(27,75,117,60)(28,76,118,61)(29,77,119,62)(30,78,120,63)(31,79,121,64)(32,80,122,65)(33,81,123,66)(34,82,124,67)(35,83,125,68)(36,84,126,69), (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), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,110)(9,109)(10,144)(11,143)(12,142)(13,141)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,120)(35,119)(36,118)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73) );
G=PermutationGroup([[(1,85,127,70),(2,86,128,71),(3,87,129,72),(4,88,130,37),(5,89,131,38),(6,90,132,39),(7,91,133,40),(8,92,134,41),(9,93,135,42),(10,94,136,43),(11,95,137,44),(12,96,138,45),(13,97,139,46),(14,98,140,47),(15,99,141,48),(16,100,142,49),(17,101,143,50),(18,102,144,51),(19,103,109,52),(20,104,110,53),(21,105,111,54),(22,106,112,55),(23,107,113,56),(24,108,114,57),(25,73,115,58),(26,74,116,59),(27,75,117,60),(28,76,118,61),(29,77,119,62),(30,78,120,63),(31,79,121,64),(32,80,122,65),(33,81,123,66),(34,82,124,67),(35,83,125,68),(36,84,126,69)], [(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)], [(1,117),(2,116),(3,115),(4,114),(5,113),(6,112),(7,111),(8,110),(9,109),(10,144),(11,143),(12,142),(13,141),(14,140),(15,139),(16,138),(17,137),(18,136),(19,135),(20,134),(21,133),(22,132),(23,131),(24,130),(25,129),(26,128),(27,127),(28,126),(29,125),(30,124),(31,123),(32,122),(33,121),(34,120),(35,119),(36,118),(37,108),(38,107),(39,106),(40,105),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 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 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4○D4 | D9 | C4×S3 | D12 | D18 | C4○D12 | C4×D9 | D36 | D36⋊5C2 |
| kernel | C4×D36 | C4⋊Dic9 | D18⋊C4 | C4×C36 | C2×C4×D9 | C2×D36 | D36 | C4×C12 | C36 | C2×C12 | C18 | C42 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 3 | 2 | 3 | 4 | 4 | 9 | 4 | 12 | 12 | 12 |
Matrix representation of C4×D36 ►in GL3(𝔽37) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 36 | 0 | 0 |
| 0 | 4 | 8 |
| 0 | 29 | 12 |
| 1 | 0 | 0 |
| 0 | 6 | 20 |
| 0 | 26 | 31 |
G:=sub<GL(3,GF(37))| [6,0,0,0,6,0,0,0,6],[36,0,0,0,4,29,0,8,12],[1,0,0,0,6,26,0,20,31] >;
C4×D36 in GAP, Magma, Sage, TeX
C_4\times D_{36} % in TeX
G:=Group("C4xD36"); // GroupNames label
G:=SmallGroup(288,83);
// by ID
G=gap.SmallGroup(288,83);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^4=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations