metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊2D9, (C4×C36)⋊9C2, (C4×D9)⋊3C4, C4.22(C4×D9), C12.72(C4×S3), C36.27(C2×C4), (C4×C12).19S3, D18⋊C4.7C2, (C4×Dic9)⋊8C2, D18.3(C2×C4), (C2×C4).64D18, Dic9⋊C4⋊17C2, C9⋊1(C42⋊C2), C18.3(C4○D4), (C2×C12).338D6, C3.(C42⋊2S3), C18.3(C22×C4), Dic9.5(C2×C4), C6.73(C4○D12), (C2×C36).71C22, (C2×C18).13C23, C2.2(D36⋊5C2), C22.10(C22×D9), (C2×Dic9).21C22, (C22×D9).14C22, C2.5(C2×C4×D9), C6.42(S3×C2×C4), (C2×C4×D9).7C2, (C2×C6).170(C22×S3), SmallGroup(288,82)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊2D9
G = < a,b,c,d | a4=b4=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 448 in 114 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, C23, C9, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, D9, C18, C18, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C42⋊C2, Dic9, Dic9, C36, C36, D18, D18, C2×C18, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C4×D9, C2×Dic9, C2×Dic9, C2×C36, C2×C36, C22×D9, C42⋊2S3, C4×Dic9, Dic9⋊C4, D18⋊C4, C4×C36, C2×C4×D9, C42⋊2D9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, D9, C4×S3, C22×S3, C42⋊C2, D18, S3×C2×C4, C4○D12, C4×D9, C22×D9, C42⋊2S3, C2×C4×D9, D36⋊5C2, C42⋊2D9
►On 144 points
Generators in S144
(1 86 14 77)(2 87 15 78)(3 88 16 79)(4 89 17 80)(5 90 18 81)(6 82 10 73)(7 83 11 74)(8 84 12 75)(9 85 13 76)(19 100 28 91)(20 101 29 92)(21 102 30 93)(22 103 31 94)(23 104 32 95)(24 105 33 96)(25 106 34 97)(26 107 35 98)(27 108 36 99)(37 118 46 109)(38 119 47 110)(39 120 48 111)(40 121 49 112)(41 122 50 113)(42 123 51 114)(43 124 52 115)(44 125 53 116)(45 126 54 117)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 139 67 130)(59 140 68 131)(60 141 69 132)(61 142 70 133)(62 143 71 134)(63 144 72 135)
(1 59 23 41)(2 60 24 42)(3 61 25 43)(4 62 26 44)(5 63 27 45)(6 55 19 37)(7 56 20 38)(8 57 21 39)(9 58 22 40)(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 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 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)(73 80)(74 79)(75 78)(76 77)(82 89)(83 88)(84 87)(85 86)(91 98)(92 97)(93 96)(94 95)(100 107)(101 106)(102 105)(103 104)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(117 126)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(135 144)
G:=sub<Sym(144)| (1,86,14,77)(2,87,15,78)(3,88,16,79)(4,89,17,80)(5,90,18,81)(6,82,10,73)(7,83,11,74)(8,84,12,75)(9,85,13,76)(19,100,28,91)(20,101,29,92)(21,102,30,93)(22,103,31,94)(23,104,32,95)(24,105,33,96)(25,106,34,97)(26,107,35,98)(27,108,36,99)(37,118,46,109)(38,119,47,110)(39,120,48,111)(40,121,49,112)(41,122,50,113)(42,123,51,114)(43,124,52,115)(44,125,53,116)(45,126,54,117)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,139,67,130)(59,140,68,131)(60,141,69,132)(61,142,70,133)(62,143,71,134)(63,144,72,135), (1,59,23,41)(2,60,24,42)(3,61,25,43)(4,62,26,44)(5,63,27,45)(6,55,19,37)(7,56,20,38)(8,57,21,39)(9,58,22,40)(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), (1,9)(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)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)(73,80)(74,79)(75,78)(76,77)(82,89)(83,88)(84,87)(85,86)(91,98)(92,97)(93,96)(94,95)(100,107)(101,106)(102,105)(103,104)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(117,126)(127,143)(128,142)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(135,144)>;
G:=Group( (1,86,14,77)(2,87,15,78)(3,88,16,79)(4,89,17,80)(5,90,18,81)(6,82,10,73)(7,83,11,74)(8,84,12,75)(9,85,13,76)(19,100,28,91)(20,101,29,92)(21,102,30,93)(22,103,31,94)(23,104,32,95)(24,105,33,96)(25,106,34,97)(26,107,35,98)(27,108,36,99)(37,118,46,109)(38,119,47,110)(39,120,48,111)(40,121,49,112)(41,122,50,113)(42,123,51,114)(43,124,52,115)(44,125,53,116)(45,126,54,117)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,139,67,130)(59,140,68,131)(60,141,69,132)(61,142,70,133)(62,143,71,134)(63,144,72,135), (1,59,23,41)(2,60,24,42)(3,61,25,43)(4,62,26,44)(5,63,27,45)(6,55,19,37)(7,56,20,38)(8,57,21,39)(9,58,22,40)(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), (1,9)(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)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)(73,80)(74,79)(75,78)(76,77)(82,89)(83,88)(84,87)(85,86)(91,98)(92,97)(93,96)(94,95)(100,107)(101,106)(102,105)(103,104)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(117,126)(127,143)(128,142)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(135,144) );
G=PermutationGroup([[(1,86,14,77),(2,87,15,78),(3,88,16,79),(4,89,17,80),(5,90,18,81),(6,82,10,73),(7,83,11,74),(8,84,12,75),(9,85,13,76),(19,100,28,91),(20,101,29,92),(21,102,30,93),(22,103,31,94),(23,104,32,95),(24,105,33,96),(25,106,34,97),(26,107,35,98),(27,108,36,99),(37,118,46,109),(38,119,47,110),(39,120,48,111),(40,121,49,112),(41,122,50,113),(42,123,51,114),(43,124,52,115),(44,125,53,116),(45,126,54,117),(55,136,64,127),(56,137,65,128),(57,138,66,129),(58,139,67,130),(59,140,68,131),(60,141,69,132),(61,142,70,133),(62,143,71,134),(63,144,72,135)], [(1,59,23,41),(2,60,24,42),(3,61,25,43),(4,62,26,44),(5,63,27,45),(6,55,19,37),(7,56,20,38),(8,57,21,39),(9,58,22,40),(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)], [(1,9),(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),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(45,54),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(63,72),(73,80),(74,79),(75,78),(76,77),(82,89),(83,88),(84,87),(85,86),(91,98),(92,97),(93,96),(94,95),(100,107),(101,106),(102,105),(103,104),(109,125),(110,124),(111,123),(112,122),(113,121),(114,120),(115,119),(116,118),(117,126),(127,143),(128,142),(129,141),(130,140),(131,139),(132,138),(133,137),(134,136),(135,144)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | C4○D4 | D9 | C4×S3 | D18 | C4○D12 | C4×D9 | D36⋊5C2 |
| kernel | C42⋊2D9 | C4×Dic9 | Dic9⋊C4 | D18⋊C4 | C4×C36 | C2×C4×D9 | C4×D9 | C4×C12 | C2×C12 | C18 | C42 | C12 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 1 | 3 | 4 | 3 | 4 | 9 | 8 | 12 | 24 |
Matrix representation of C42⋊2D9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 5 | 10 |
| 0 | 0 | 27 | 32 |
| 31 | 20 | 0 | 0 |
| 17 | 11 | 0 | 0 |
| 0 | 0 | 31 | 20 |
| 0 | 0 | 17 | 11 |
| 31 | 20 | 0 | 0 |
| 26 | 6 | 0 | 0 |
| 0 | 0 | 31 | 20 |
| 0 | 0 | 26 | 6 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[6,0,0,0,0,6,0,0,0,0,5,27,0,0,10,32],[31,17,0,0,20,11,0,0,0,0,31,17,0,0,20,11],[31,26,0,0,20,6,0,0,0,0,31,26,0,0,20,6] >;
C42⋊2D9 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2D_9
% in TeX
G:=Group("C4^2:2D9"); // GroupNames label
G:=SmallGroup(288,82);
// by ID
G=gap.SmallGroup(288,82);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations