metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊6D9, C4⋊Dic9⋊7C2, D18⋊C4.4C2, (C2×C4).11D18, Dic9⋊C4⋊13C2, C9⋊3(C42⋊2C2), (C4×Dic9)⋊14C2, (C2×C12).183D6, C6.84(C4○D12), C18.14(C4○D4), (C2×C36).14C22, (C2×C18).39C23, C2.7(Q8⋊3D9), C6.84(D4⋊2S3), C2.14(D4⋊2D9), C6.42(Q8⋊3S3), C2.16(D36⋊5C2), (C22×D9).8C22, C22.53(C22×D9), (C2×Dic9).35C22, (C9×C4⋊C4)⋊9C2, C3.(C4⋊C4⋊S3), (C3×C4⋊C4).16S3, (C2×C6).196(C22×S3), SmallGroup(288,108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊D9
G = < a,b,c,d | a4=b4=c9=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 396 in 90 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, D9, C18, C2×Dic3, C2×C12, C22×S3, C42⋊2C2, Dic9, C36, D18, C2×C18, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic9, C2×C36, C22×D9, C4⋊C4⋊S3, C4×Dic9, Dic9⋊C4, C4⋊Dic9, D18⋊C4, C9×C4⋊C4, C4⋊C4⋊D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, C42⋊2C2, D18, C4○D12, D4⋊2S3, Q8⋊3S3, C22×D9, C4⋊C4⋊S3, D36⋊5C2, D4⋊2D9, Q8⋊3D9, C4⋊C4⋊D9
►On 144 points
Generators in S144
(1 68 14 59)(2 69 15 60)(3 70 16 61)(4 71 17 62)(5 72 18 63)(6 64 10 55)(7 65 11 56)(8 66 12 57)(9 67 13 58)(19 46 28 37)(20 47 29 38)(21 48 30 39)(22 49 31 40)(23 50 32 41)(24 51 33 42)(25 52 34 43)(26 53 35 44)(27 54 36 45)(73 127 82 136)(74 128 83 137)(75 129 84 138)(76 130 85 139)(77 131 86 140)(78 132 87 141)(79 133 88 142)(80 134 89 143)(81 135 90 144)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)
(1 95 23 77)(2 96 24 78)(3 97 25 79)(4 98 26 80)(5 99 27 81)(6 91 19 73)(7 92 20 74)(8 93 21 75)(9 94 22 76)(10 100 28 82)(11 101 29 83)(12 102 30 84)(13 103 31 85)(14 104 32 86)(15 105 33 87)(16 106 34 88)(17 107 35 89)(18 108 36 90)(37 127 55 109)(38 128 56 110)(39 129 57 111)(40 130 58 112)(41 131 59 113)(42 132 60 114)(43 133 61 115)(44 134 62 116)(45 135 63 117)(46 136 64 118)(47 137 65 119)(48 138 66 120)(49 139 67 121)(50 140 68 122)(51 141 69 123)(52 142 70 124)(53 143 71 125)(54 144 72 126)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(81 90)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(99 108)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 144)(118 134)(119 133)(120 132)(121 131)(122 130)(123 129)(124 128)(125 127)(126 135)
G:=sub<Sym(144)| (1,68,14,59)(2,69,15,60)(3,70,16,61)(4,71,17,62)(5,72,18,63)(6,64,10,55)(7,65,11,56)(8,66,12,57)(9,67,13,58)(19,46,28,37)(20,47,29,38)(21,48,30,39)(22,49,31,40)(23,50,32,41)(24,51,33,42)(25,52,34,43)(26,53,35,44)(27,54,36,45)(73,127,82,136)(74,128,83,137)(75,129,84,138)(76,130,85,139)(77,131,86,140)(78,132,87,141)(79,133,88,142)(80,134,89,143)(81,135,90,144)(91,109,100,118)(92,110,101,119)(93,111,102,120)(94,112,103,121)(95,113,104,122)(96,114,105,123)(97,115,106,124)(98,116,107,125)(99,117,108,126), (1,95,23,77)(2,96,24,78)(3,97,25,79)(4,98,26,80)(5,99,27,81)(6,91,19,73)(7,92,20,74)(8,93,21,75)(9,94,22,76)(10,100,28,82)(11,101,29,83)(12,102,30,84)(13,103,31,85)(14,104,32,86)(15,105,33,87)(16,106,34,88)(17,107,35,89)(18,108,36,90)(37,127,55,109)(38,128,56,110)(39,129,57,111)(40,130,58,112)(41,131,59,113)(42,132,60,114)(43,133,61,115)(44,134,62,116)(45,135,63,117)(46,136,64,118)(47,137,65,119)(48,138,66,120)(49,139,67,121)(50,140,68,122)(51,141,69,123)(52,142,70,124)(53,143,71,125)(54,144,72,126), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(81,90)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(99,108)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,144)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(126,135)>;
G:=Group( (1,68,14,59)(2,69,15,60)(3,70,16,61)(4,71,17,62)(5,72,18,63)(6,64,10,55)(7,65,11,56)(8,66,12,57)(9,67,13,58)(19,46,28,37)(20,47,29,38)(21,48,30,39)(22,49,31,40)(23,50,32,41)(24,51,33,42)(25,52,34,43)(26,53,35,44)(27,54,36,45)(73,127,82,136)(74,128,83,137)(75,129,84,138)(76,130,85,139)(77,131,86,140)(78,132,87,141)(79,133,88,142)(80,134,89,143)(81,135,90,144)(91,109,100,118)(92,110,101,119)(93,111,102,120)(94,112,103,121)(95,113,104,122)(96,114,105,123)(97,115,106,124)(98,116,107,125)(99,117,108,126), (1,95,23,77)(2,96,24,78)(3,97,25,79)(4,98,26,80)(5,99,27,81)(6,91,19,73)(7,92,20,74)(8,93,21,75)(9,94,22,76)(10,100,28,82)(11,101,29,83)(12,102,30,84)(13,103,31,85)(14,104,32,86)(15,105,33,87)(16,106,34,88)(17,107,35,89)(18,108,36,90)(37,127,55,109)(38,128,56,110)(39,129,57,111)(40,130,58,112)(41,131,59,113)(42,132,60,114)(43,133,61,115)(44,134,62,116)(45,135,63,117)(46,136,64,118)(47,137,65,119)(48,138,66,120)(49,139,67,121)(50,140,68,122)(51,141,69,123)(52,142,70,124)(53,143,71,125)(54,144,72,126), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(81,90)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(99,108)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,144)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(126,135) );
G=PermutationGroup([[(1,68,14,59),(2,69,15,60),(3,70,16,61),(4,71,17,62),(5,72,18,63),(6,64,10,55),(7,65,11,56),(8,66,12,57),(9,67,13,58),(19,46,28,37),(20,47,29,38),(21,48,30,39),(22,49,31,40),(23,50,32,41),(24,51,33,42),(25,52,34,43),(26,53,35,44),(27,54,36,45),(73,127,82,136),(74,128,83,137),(75,129,84,138),(76,130,85,139),(77,131,86,140),(78,132,87,141),(79,133,88,142),(80,134,89,143),(81,135,90,144),(91,109,100,118),(92,110,101,119),(93,111,102,120),(94,112,103,121),(95,113,104,122),(96,114,105,123),(97,115,106,124),(98,116,107,125),(99,117,108,126)], [(1,95,23,77),(2,96,24,78),(3,97,25,79),(4,98,26,80),(5,99,27,81),(6,91,19,73),(7,92,20,74),(8,93,21,75),(9,94,22,76),(10,100,28,82),(11,101,29,83),(12,102,30,84),(13,103,31,85),(14,104,32,86),(15,105,33,87),(16,106,34,88),(17,107,35,89),(18,108,36,90),(37,127,55,109),(38,128,56,110),(39,129,57,111),(40,130,58,112),(41,131,59,113),(42,132,60,114),(43,133,61,115),(44,134,62,116),(45,135,63,117),(46,136,64,118),(47,137,65,119),(48,138,66,120),(49,139,67,121),(50,140,68,122),(51,141,69,123),(52,142,70,124),(53,143,71,125),(54,144,72,126)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,72),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(81,90),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(99,108),(109,143),(110,142),(111,141),(112,140),(113,139),(114,138),(115,137),(116,136),(117,144),(118,134),(119,133),(120,132),(121,131),(122,130),(123,129),(124,128),(125,127),(126,135)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 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 | 36 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | C4○D4 | D9 | D18 | C4○D12 | D36⋊5C2 | D4⋊2S3 | Q8⋊3S3 | D4⋊2D9 | Q8⋊3D9 |
| kernel | C4⋊C4⋊D9 | C4×Dic9 | Dic9⋊C4 | C4⋊Dic9 | D18⋊C4 | C9×C4⋊C4 | C3×C4⋊C4 | C2×C12 | C18 | C4⋊C4 | C2×C4 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 3 | 6 | 3 | 9 | 4 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of C4⋊C4⋊D9 ►in GL4(𝔽37) generated by
| 30 | 23 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 6 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 31 |
| 0 | 0 | 6 | 0 |
| 31 | 20 | 0 | 0 |
| 17 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 31 | 20 | 0 | 0 |
| 26 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 36 |
G:=sub<GL(4,GF(37))| [30,14,0,0,23,7,0,0,0,0,31,0,0,0,0,6],[6,0,0,0,0,6,0,0,0,0,0,6,0,0,31,0],[31,17,0,0,20,11,0,0,0,0,1,0,0,0,0,1],[31,26,0,0,20,6,0,0,0,0,1,0,0,0,0,36] >;
C4⋊C4⋊D9 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes D_9
% in TeX
G:=Group("C4:C4:D9"); // GroupNames label
G:=SmallGroup(288,108);
// by ID
G=gap.SmallGroup(288,108);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,254,219,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^2=1,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations