metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊3D9, (C4×C36)⋊1C2, (C4×C12).2S3, Dic9⋊C4⋊1C2, D18⋊C4.1C2, (C2×C4).66D18, C9⋊1(C42⋊2C2), C18.6(C4○D4), (C2×C12).340D6, C3.(C42⋊3S3), C6.76(C4○D12), (C2×C36).73C22, (C2×C18).17C23, C2.8(D36⋊5C2), (C2×Dic9).4C22, (C22×D9).3C22, C22.38(C22×D9), (C2×C6).174(C22×S3), SmallGroup(288,86)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊3D9
G = < a,b,c,d | a4=b4=c9=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >
Subgroups: 396 in 90 conjugacy classes, 38 normal (11 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, D9, C18, C2×Dic3, C2×C12, C22×S3, C42⋊2C2, Dic9, C36, D18, C2×C18, Dic3⋊C4, D6⋊C4, C4×C12, C2×Dic9, C2×C36, C22×D9, C42⋊3S3, Dic9⋊C4, D18⋊C4, C4×C36, C42⋊3D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, C42⋊2C2, D18, C4○D12, C22×D9, C42⋊3S3, D36⋊5C2, C42⋊3D9
►On 144 points
Generators in S144
(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 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)(91 136 100 127)(92 137 101 128)(93 138 102 129)(94 139 103 130)(95 140 104 131)(96 141 105 132)(97 142 106 133)(98 143 107 134)(99 144 108 135)
(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 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 72)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(54 63)(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)(109 134)(110 133)(111 132)(112 131)(113 130)(114 129)(115 128)(116 127)(117 135)(118 143)(119 142)(120 141)(121 140)(122 139)(123 138)(124 137)(125 136)(126 144)
G:=sub<Sym(144)| (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), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(91,136,100,127)(92,137,101,128)(93,138,102,129)(94,139,103,130)(95,140,104,131)(96,141,105,132)(97,142,106,133)(98,143,107,134)(99,144,108,135), (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)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,72)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(54,63)(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)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,135)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)(125,136)(126,144)>;
G:=Group( (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), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(91,136,100,127)(92,137,101,128)(93,138,102,129)(94,139,103,130)(95,140,104,131)(96,141,105,132)(97,142,106,133)(98,143,107,134)(99,144,108,135), (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)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,72)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(54,63)(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)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,135)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)(125,136)(126,144) );
G=PermutationGroup([[(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)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63),(73,118,82,109),(74,119,83,110),(75,120,84,111),(76,121,85,112),(77,122,86,113),(78,123,87,114),(79,124,88,115),(80,125,89,116),(81,126,90,117),(91,136,100,127),(92,137,101,128),(93,138,102,129),(94,139,103,130),(95,140,104,131),(96,141,105,132),(97,142,106,133),(98,143,107,134),(99,144,108,135)], [(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),(37,71),(38,70),(39,69),(40,68),(41,67),(42,66),(43,65),(44,64),(45,72),(46,62),(47,61),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(54,63),(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),(109,134),(110,133),(111,132),(112,131),(113,130),(114,129),(115,128),(116,127),(117,135),(118,143),(119,142),(120,141),(121,140),(122,139),(123,138),(124,137),(125,136),(126,144)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 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 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D6 | C4○D4 | D9 | D18 | C4○D12 | D36⋊5C2 |
| kernel | C42⋊3D9 | Dic9⋊C4 | D18⋊C4 | C4×C36 | C4×C12 | C2×C12 | C18 | C42 | C2×C4 | C6 | C2 |
| # reps | 1 | 3 | 3 | 1 | 1 | 3 | 6 | 3 | 9 | 12 | 36 |
Matrix representation of C42⋊3D9 ►in GL4(𝔽37) generated by
| 5 | 27 | 0 | 0 |
| 10 | 32 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 31 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 7 | 23 |
| 0 | 0 | 14 | 30 |
| 36 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 11 | 20 |
| 0 | 0 | 17 | 31 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 20 | 6 |
| 0 | 0 | 26 | 17 |
G:=sub<GL(4,GF(37))| [5,10,0,0,27,32,0,0,0,0,6,0,0,0,0,6],[31,0,0,0,0,31,0,0,0,0,7,14,0,0,23,30],[36,36,0,0,1,0,0,0,0,0,11,17,0,0,20,31],[0,1,0,0,1,0,0,0,0,0,20,26,0,0,6,17] >;
C42⋊3D9 in GAP, Magma, Sage, TeX
C_4^2\rtimes_3D_9
% in TeX
G:=Group("C4^2:3D9"); // GroupNames label
G:=SmallGroup(288,86);
// by ID
G=gap.SmallGroup(288,86);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations