direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C9⋊D4, C36⋊8D4, C23.27D18, C9⋊4(C4×D4), D18⋊4(C2×C4), C22⋊3(C4×D9), (C22×C4)⋊4D9, D18⋊C4⋊18C2, (C22×C36)⋊9C2, Dic9⋊2(C2×C4), C18.42(C2×D4), Dic9⋊C4⋊18C2, (C4×Dic9)⋊16C2, (C2×C4).103D18, (C2×C12).344D6, C6.87(C4○D12), C18.17(C4○D4), (C2×C18).46C23, C18.20(C22×C4), (C22×C12).34S3, (C2×C36).76C22, (C22×C6).140D6, C18.D4⋊14C2, C2.5(D36⋊5C2), C12.127(C3⋊D4), C22.24(C22×D9), (C22×C18).38C22, (C2×Dic9).39C22, (C22×D9).24C22, C3.(C4×C3⋊D4), (C2×C4×D9)⋊14C2, C6.59(S3×C2×C4), C2.20(C2×C4×D9), (C2×C18)⋊5(C2×C4), C2.3(C2×C9⋊D4), (C2×C6).45(C4×S3), (C2×C9⋊D4).7C2, C6.89(C2×C3⋊D4), (C2×C6).203(C22×S3), SmallGroup(288,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C9⋊D4
G = < a,b,c,d | a4=b9=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 536 in 141 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, D9, C18, C18, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×D4, Dic9, Dic9, C36, C36, D18, D18, C2×C18, C2×C18, C2×C18, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C4×D9, C2×Dic9, C9⋊D4, C2×C36, C2×C36, C22×D9, C22×C18, C4×C3⋊D4, C4×Dic9, Dic9⋊C4, D18⋊C4, C18.D4, C2×C4×D9, C2×C9⋊D4, C22×C36, C4×C9⋊D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, D9, C4×S3, C3⋊D4, C22×S3, C4×D4, D18, S3×C2×C4, C4○D12, C2×C3⋊D4, C4×D9, C9⋊D4, C22×D9, C4×C3⋊D4, C2×C4×D9, D36⋊5C2, C2×C9⋊D4, C4×C9⋊D4
►On 144 points
Generators in S144
(1 58 22 40)(2 59 23 41)(3 60 24 42)(4 61 25 43)(5 62 26 44)(6 63 27 45)(7 55 19 37)(8 56 20 38)(9 57 21 39)(10 64 28 46)(11 65 29 47)(12 66 30 48)(13 67 31 49)(14 68 32 50)(15 69 33 51)(16 70 34 52)(17 71 35 53)(18 72 36 54)(73 127 91 109)(74 128 92 110)(75 129 93 111)(76 130 94 112)(77 131 95 113)(78 132 96 114)(79 133 97 115)(80 134 98 116)(81 135 99 117)(82 136 100 118)(83 137 101 119)(84 138 102 120)(85 139 103 121)(86 140 104 122)(87 141 105 123)(88 142 106 124)(89 143 107 125)(90 144 108 126)
(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 13 103)(2 93 14 102)(3 92 15 101)(4 91 16 100)(5 99 17 108)(6 98 18 107)(7 97 10 106)(8 96 11 105)(9 95 12 104)(19 79 28 88)(20 78 29 87)(21 77 30 86)(22 76 31 85)(23 75 32 84)(24 74 33 83)(25 73 34 82)(26 81 35 90)(27 80 36 89)(37 133 46 142)(38 132 47 141)(39 131 48 140)(40 130 49 139)(41 129 50 138)(42 128 51 137)(43 127 52 136)(44 135 53 144)(45 134 54 143)(55 115 64 124)(56 114 65 123)(57 113 66 122)(58 112 67 121)(59 111 68 120)(60 110 69 119)(61 109 70 118)(62 117 71 126)(63 116 72 125)
(2 9)(3 8)(4 7)(5 6)(10 16)(11 15)(12 14)(17 18)(19 25)(20 24)(21 23)(26 27)(28 34)(29 33)(30 32)(35 36)(37 43)(38 42)(39 41)(44 45)(46 52)(47 51)(48 50)(53 54)(55 61)(56 60)(57 59)(62 63)(64 70)(65 69)(66 68)(71 72)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 90)(81 89)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 108)(99 107)(109 124)(110 123)(111 122)(112 121)(113 120)(114 119)(115 118)(116 126)(117 125)(127 142)(128 141)(129 140)(130 139)(131 138)(132 137)(133 136)(134 144)(135 143)
G:=sub<Sym(144)| (1,58,22,40)(2,59,23,41)(3,60,24,42)(4,61,25,43)(5,62,26,44)(6,63,27,45)(7,55,19,37)(8,56,20,38)(9,57,21,39)(10,64,28,46)(11,65,29,47)(12,66,30,48)(13,67,31,49)(14,68,32,50)(15,69,33,51)(16,70,34,52)(17,71,35,53)(18,72,36,54)(73,127,91,109)(74,128,92,110)(75,129,93,111)(76,130,94,112)(77,131,95,113)(78,132,96,114)(79,133,97,115)(80,134,98,116)(81,135,99,117)(82,136,100,118)(83,137,101,119)(84,138,102,120)(85,139,103,121)(86,140,104,122)(87,141,105,123)(88,142,106,124)(89,143,107,125)(90,144,108,126), (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), (1,94,13,103)(2,93,14,102)(3,92,15,101)(4,91,16,100)(5,99,17,108)(6,98,18,107)(7,97,10,106)(8,96,11,105)(9,95,12,104)(19,79,28,88)(20,78,29,87)(21,77,30,86)(22,76,31,85)(23,75,32,84)(24,74,33,83)(25,73,34,82)(26,81,35,90)(27,80,36,89)(37,133,46,142)(38,132,47,141)(39,131,48,140)(40,130,49,139)(41,129,50,138)(42,128,51,137)(43,127,52,136)(44,135,53,144)(45,134,54,143)(55,115,64,124)(56,114,65,123)(57,113,66,122)(58,112,67,121)(59,111,68,120)(60,110,69,119)(61,109,70,118)(62,117,71,126)(63,116,72,125), (2,9)(3,8)(4,7)(5,6)(10,16)(11,15)(12,14)(17,18)(19,25)(20,24)(21,23)(26,27)(28,34)(29,33)(30,32)(35,36)(37,43)(38,42)(39,41)(44,45)(46,52)(47,51)(48,50)(53,54)(55,61)(56,60)(57,59)(62,63)(64,70)(65,69)(66,68)(71,72)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,90)(81,89)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,108)(99,107)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,126)(117,125)(127,142)(128,141)(129,140)(130,139)(131,138)(132,137)(133,136)(134,144)(135,143)>;
G:=Group( (1,58,22,40)(2,59,23,41)(3,60,24,42)(4,61,25,43)(5,62,26,44)(6,63,27,45)(7,55,19,37)(8,56,20,38)(9,57,21,39)(10,64,28,46)(11,65,29,47)(12,66,30,48)(13,67,31,49)(14,68,32,50)(15,69,33,51)(16,70,34,52)(17,71,35,53)(18,72,36,54)(73,127,91,109)(74,128,92,110)(75,129,93,111)(76,130,94,112)(77,131,95,113)(78,132,96,114)(79,133,97,115)(80,134,98,116)(81,135,99,117)(82,136,100,118)(83,137,101,119)(84,138,102,120)(85,139,103,121)(86,140,104,122)(87,141,105,123)(88,142,106,124)(89,143,107,125)(90,144,108,126), (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), (1,94,13,103)(2,93,14,102)(3,92,15,101)(4,91,16,100)(5,99,17,108)(6,98,18,107)(7,97,10,106)(8,96,11,105)(9,95,12,104)(19,79,28,88)(20,78,29,87)(21,77,30,86)(22,76,31,85)(23,75,32,84)(24,74,33,83)(25,73,34,82)(26,81,35,90)(27,80,36,89)(37,133,46,142)(38,132,47,141)(39,131,48,140)(40,130,49,139)(41,129,50,138)(42,128,51,137)(43,127,52,136)(44,135,53,144)(45,134,54,143)(55,115,64,124)(56,114,65,123)(57,113,66,122)(58,112,67,121)(59,111,68,120)(60,110,69,119)(61,109,70,118)(62,117,71,126)(63,116,72,125), (2,9)(3,8)(4,7)(5,6)(10,16)(11,15)(12,14)(17,18)(19,25)(20,24)(21,23)(26,27)(28,34)(29,33)(30,32)(35,36)(37,43)(38,42)(39,41)(44,45)(46,52)(47,51)(48,50)(53,54)(55,61)(56,60)(57,59)(62,63)(64,70)(65,69)(66,68)(71,72)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,90)(81,89)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,108)(99,107)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,126)(117,125)(127,142)(128,141)(129,140)(130,139)(131,138)(132,137)(133,136)(134,144)(135,143) );
G=PermutationGroup([[(1,58,22,40),(2,59,23,41),(3,60,24,42),(4,61,25,43),(5,62,26,44),(6,63,27,45),(7,55,19,37),(8,56,20,38),(9,57,21,39),(10,64,28,46),(11,65,29,47),(12,66,30,48),(13,67,31,49),(14,68,32,50),(15,69,33,51),(16,70,34,52),(17,71,35,53),(18,72,36,54),(73,127,91,109),(74,128,92,110),(75,129,93,111),(76,130,94,112),(77,131,95,113),(78,132,96,114),(79,133,97,115),(80,134,98,116),(81,135,99,117),(82,136,100,118),(83,137,101,119),(84,138,102,120),(85,139,103,121),(86,140,104,122),(87,141,105,123),(88,142,106,124),(89,143,107,125),(90,144,108,126)], [(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)], [(1,94,13,103),(2,93,14,102),(3,92,15,101),(4,91,16,100),(5,99,17,108),(6,98,18,107),(7,97,10,106),(8,96,11,105),(9,95,12,104),(19,79,28,88),(20,78,29,87),(21,77,30,86),(22,76,31,85),(23,75,32,84),(24,74,33,83),(25,73,34,82),(26,81,35,90),(27,80,36,89),(37,133,46,142),(38,132,47,141),(39,131,48,140),(40,130,49,139),(41,129,50,138),(42,128,51,137),(43,127,52,136),(44,135,53,144),(45,134,54,143),(55,115,64,124),(56,114,65,123),(57,113,66,122),(58,112,67,121),(59,111,68,120),(60,110,69,119),(61,109,70,118),(62,117,71,126),(63,116,72,125)], [(2,9),(3,8),(4,7),(5,6),(10,16),(11,15),(12,14),(17,18),(19,25),(20,24),(21,23),(26,27),(28,34),(29,33),(30,32),(35,36),(37,43),(38,42),(39,41),(44,45),(46,52),(47,51),(48,50),(53,54),(55,61),(56,60),(57,59),(62,63),(64,70),(65,69),(66,68),(71,72),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,90),(81,89),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,108),(99,107),(109,124),(110,123),(111,122),(112,121),(113,120),(114,119),(115,118),(116,126),(117,125),(127,142),(128,141),(129,140),(130,139),(131,138),(132,137),(133,136),(134,144),(135,143)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | ··· | 6G | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | C4×S3 | D18 | D18 | C4○D12 | C9⋊D4 | C4×D9 | D36⋊5C2 |
| kernel | C4×C9⋊D4 | C4×Dic9 | Dic9⋊C4 | D18⋊C4 | C18.D4 | C2×C4×D9 | C2×C9⋊D4 | C22×C36 | C9⋊D4 | C22×C12 | C36 | C2×C12 | C22×C6 | C18 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 2 | 3 | 4 | 4 | 6 | 3 | 4 | 12 | 12 | 12 |
Matrix representation of C4×C9⋊D4 ►in GL4(𝔽37) generated by
| 31 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 36 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 26 |
| 0 | 0 | 11 | 17 |
| 7 | 14 | 0 | 0 |
| 7 | 30 | 0 | 0 |
| 0 | 0 | 30 | 23 |
| 0 | 0 | 30 | 7 |
| 1 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 36 | 36 |
G:=sub<GL(4,GF(37))| [31,0,0,0,0,31,0,0,0,0,36,0,0,0,0,36],[36,1,0,0,36,0,0,0,0,0,6,11,0,0,26,17],[7,7,0,0,14,30,0,0,0,0,30,30,0,0,23,7],[1,36,0,0,0,36,0,0,0,0,1,36,0,0,0,36] >;
C4×C9⋊D4 in GAP, Magma, Sage, TeX
C_4\times C_9\rtimes D_4
% in TeX
G:=Group("C4xC9:D4"); // GroupNames label
G:=SmallGroup(288,138);
// by ID
G=gap.SmallGroup(288,138);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^9=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations