metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊1D36, C36⋊4D4, C42⋊6D9, C12.38D12, (C4×C36)⋊4C2, (C2×D36)⋊1C2, C9⋊1(C4⋊1D4), (C4×C12).6S3, C18.3(C2×D4), C2.5(C2×D36), C3.(C4⋊D12), C6.32(C2×D12), (C2×C4).78D18, (C2×C12).369D6, (C2×C18).15C23, (C2×C36).87C22, (C22×D9).1C22, C22.36(C22×D9), (C2×C6).172(C22×S3), SmallGroup(288,84)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊6D9
G = < a,b,c | a36=b4=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 1020 in 162 conjugacy classes, 56 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C9, C12, D6, C2×C6, C42, C2×D4, D9, C18, D12, C2×C12, C22×S3, C4⋊1D4, C36, D18, C2×C18, C4×C12, C2×D12, D36, C2×C36, C22×D9, C4⋊D12, C4×C36, C2×D36, C42⋊6D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C22×S3, C4⋊1D4, D18, C2×D12, D36, C22×D9, C4⋊D12, C2×D36, C42⋊6D9
►On 144 points
Generators in S144
(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 93 48 118)(2 94 49 119)(3 95 50 120)(4 96 51 121)(5 97 52 122)(6 98 53 123)(7 99 54 124)(8 100 55 125)(9 101 56 126)(10 102 57 127)(11 103 58 128)(12 104 59 129)(13 105 60 130)(14 106 61 131)(15 107 62 132)(16 108 63 133)(17 73 64 134)(18 74 65 135)(19 75 66 136)(20 76 67 137)(21 77 68 138)(22 78 69 139)(23 79 70 140)(24 80 71 141)(25 81 72 142)(26 82 37 143)(27 83 38 144)(28 84 39 109)(29 85 40 110)(30 86 41 111)(31 87 42 112)(32 88 43 113)(33 89 44 114)(34 90 45 115)(35 91 46 116)(36 92 47 117)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(73 77)(74 76)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(109 127)(110 126)(111 125)(112 124)(113 123)(114 122)(115 121)(116 120)(117 119)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)(134 138)(135 137)
G:=sub<Sym(144)| (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), (1,93,48,118)(2,94,49,119)(3,95,50,120)(4,96,51,121)(5,97,52,122)(6,98,53,123)(7,99,54,124)(8,100,55,125)(9,101,56,126)(10,102,57,127)(11,103,58,128)(12,104,59,129)(13,105,60,130)(14,106,61,131)(15,107,62,132)(16,108,63,133)(17,73,64,134)(18,74,65,135)(19,75,66,136)(20,76,67,137)(21,77,68,138)(22,78,69,139)(23,79,70,140)(24,80,71,141)(25,81,72,142)(26,82,37,143)(27,83,38,144)(28,84,39,109)(29,85,40,110)(30,86,41,111)(31,87,42,112)(32,88,43,113)(33,89,44,114)(34,90,45,115)(35,91,46,116)(36,92,47,117), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(73,77)(74,76)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(109,127)(110,126)(111,125)(112,124)(113,123)(114,122)(115,121)(116,120)(117,119)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)>;
G:=Group( (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), (1,93,48,118)(2,94,49,119)(3,95,50,120)(4,96,51,121)(5,97,52,122)(6,98,53,123)(7,99,54,124)(8,100,55,125)(9,101,56,126)(10,102,57,127)(11,103,58,128)(12,104,59,129)(13,105,60,130)(14,106,61,131)(15,107,62,132)(16,108,63,133)(17,73,64,134)(18,74,65,135)(19,75,66,136)(20,76,67,137)(21,77,68,138)(22,78,69,139)(23,79,70,140)(24,80,71,141)(25,81,72,142)(26,82,37,143)(27,83,38,144)(28,84,39,109)(29,85,40,110)(30,86,41,111)(31,87,42,112)(32,88,43,113)(33,89,44,114)(34,90,45,115)(35,91,46,116)(36,92,47,117), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(73,77)(74,76)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(109,127)(110,126)(111,125)(112,124)(113,123)(114,122)(115,121)(116,120)(117,119)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137) );
G=PermutationGroup([[(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)], [(1,93,48,118),(2,94,49,119),(3,95,50,120),(4,96,51,121),(5,97,52,122),(6,98,53,123),(7,99,54,124),(8,100,55,125),(9,101,56,126),(10,102,57,127),(11,103,58,128),(12,104,59,129),(13,105,60,130),(14,106,61,131),(15,107,62,132),(16,108,63,133),(17,73,64,134),(18,74,65,135),(19,75,66,136),(20,76,67,137),(21,77,68,138),(22,78,69,139),(23,79,70,140),(24,80,71,141),(25,81,72,142),(26,82,37,143),(27,83,38,144),(28,84,39,109),(29,85,40,110),(30,86,41,111),(31,87,42,112),(32,88,43,113),(33,89,44,114),(34,90,45,115),(35,91,46,116),(36,92,47,117)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,60),(26,59),(27,58),(28,57),(29,56),(30,55),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(73,77),(74,76),(78,108),(79,107),(80,106),(81,105),(82,104),(83,103),(84,102),(85,101),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(109,127),(110,126),(111,125),(112,124),(113,123),(114,122),(115,121),(116,120),(117,119),(128,144),(129,143),(130,142),(131,141),(132,140),(133,139),(134,138),(135,137)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 36 | 36 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D9 | D12 | D18 | D36 |
| kernel | C42⋊6D9 | C4×C36 | C2×D36 | C4×C12 | C36 | C2×C12 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 1 | 6 | 1 | 6 | 3 | 3 | 12 | 9 | 36 |
Matrix representation of C42⋊6D9 ►in GL6(𝔽37)
| 27 | 32 | 0 | 0 | 0 | 0 |
| 5 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 17 |
| 0 | 0 | 0 | 0 | 20 | 6 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 27 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 3 | 0 | 0 |
| 0 | 0 | 11 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(37))| [27,5,0,0,0,0,32,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,20,0,0,0,0,17,6],[5,27,0,0,0,0,10,32,0,0,0,0,0,0,15,11,0,0,0,0,3,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,1,0,0,0,0,0,1,0,0,0,0,0,0,36,10,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C42⋊6D9 in GAP, Magma, Sage, TeX
C_4^2\rtimes_6D_9
% in TeX
G:=Group("C4^2:6D9"); // GroupNames label
G:=SmallGroup(288,84);
// by ID
G=gap.SmallGroup(288,84);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations