metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D36.2C4, C24.90D6, C8.22D18, C72.23C22, C36.37C23, Dic18.2C4, (C2×C8)⋊7D9, (C8×D9)⋊6C2, C9⋊1(C8○D4), (C2×C72)⋊10C2, C8⋊D9⋊7C2, C3.(C8○D12), C4.10(C4×D9), C9⋊D4.2C4, C12.59(C4×S3), (C2×C24).29S3, C36.20(C2×C4), D18.1(C2×C4), (C2×C4).80D18, C9⋊C8.11C22, C22.2(C4×D9), (C2×C12).393D6, C4.Dic9⋊11C2, Dic9.3(C2×C4), C4.37(C22×D9), D36⋊5C2.6C2, C18.14(C22×C4), (C2×C36).96C22, (C4×D9).15C22, C12.198(C22×S3), C6.53(S3×C2×C4), C2.15(C2×C4×D9), (C2×C6).41(C4×S3), (C2×C18).15(C2×C4), SmallGroup(288,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D36.2C4
G = < a,b,c | a36=b2=1, c4=a18, bab=a-1, ac=ca, bc=cb >
Subgroups: 332 in 93 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), C4○D4, D9, C18, C18, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, Dic9, C36, D18, C2×C18, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C9⋊C8, C72, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C8○D12, C8×D9, C8⋊D9, C4.Dic9, C2×C72, D36⋊5C2, D36.2C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, D9, C4×S3, C22×S3, C8○D4, D18, S3×C2×C4, C4×D9, C22×D9, C8○D12, C2×C4×D9, D36.2C4
►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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(70 72)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(109 133)(110 132)(111 131)(112 130)(113 129)(114 128)(115 127)(116 126)(117 125)(118 124)(119 123)(120 122)(134 144)(135 143)(136 142)(137 141)(138 140)
(1 40 102 126 19 58 84 144)(2 41 103 127 20 59 85 109)(3 42 104 128 21 60 86 110)(4 43 105 129 22 61 87 111)(5 44 106 130 23 62 88 112)(6 45 107 131 24 63 89 113)(7 46 108 132 25 64 90 114)(8 47 73 133 26 65 91 115)(9 48 74 134 27 66 92 116)(10 49 75 135 28 67 93 117)(11 50 76 136 29 68 94 118)(12 51 77 137 30 69 95 119)(13 52 78 138 31 70 96 120)(14 53 79 139 32 71 97 121)(15 54 80 140 33 72 98 122)(16 55 81 141 34 37 99 123)(17 56 82 142 35 38 100 124)(18 57 83 143 36 39 101 125)
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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(70,72)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(109,133)(110,132)(111,131)(112,130)(113,129)(114,128)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)(134,144)(135,143)(136,142)(137,141)(138,140), (1,40,102,126,19,58,84,144)(2,41,103,127,20,59,85,109)(3,42,104,128,21,60,86,110)(4,43,105,129,22,61,87,111)(5,44,106,130,23,62,88,112)(6,45,107,131,24,63,89,113)(7,46,108,132,25,64,90,114)(8,47,73,133,26,65,91,115)(9,48,74,134,27,66,92,116)(10,49,75,135,28,67,93,117)(11,50,76,136,29,68,94,118)(12,51,77,137,30,69,95,119)(13,52,78,138,31,70,96,120)(14,53,79,139,32,71,97,121)(15,54,80,140,33,72,98,122)(16,55,81,141,34,37,99,123)(17,56,82,142,35,38,100,124)(18,57,83,143,36,39,101,125)>;
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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(70,72)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(109,133)(110,132)(111,131)(112,130)(113,129)(114,128)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)(134,144)(135,143)(136,142)(137,141)(138,140), (1,40,102,126,19,58,84,144)(2,41,103,127,20,59,85,109)(3,42,104,128,21,60,86,110)(4,43,105,129,22,61,87,111)(5,44,106,130,23,62,88,112)(6,45,107,131,24,63,89,113)(7,46,108,132,25,64,90,114)(8,47,73,133,26,65,91,115)(9,48,74,134,27,66,92,116)(10,49,75,135,28,67,93,117)(11,50,76,136,29,68,94,118)(12,51,77,137,30,69,95,119)(13,52,78,138,31,70,96,120)(14,53,79,139,32,71,97,121)(15,54,80,140,33,72,98,122)(16,55,81,141,34,37,99,123)(17,56,82,142,35,38,100,124)(18,57,83,143,36,39,101,125) );
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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(28,36),(29,35),(30,34),(31,33),(37,69),(38,68),(39,67),(40,66),(41,65),(42,64),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(70,72),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98),(109,133),(110,132),(111,131),(112,130),(113,129),(114,128),(115,127),(116,126),(117,125),(118,124),(119,123),(120,122),(134,144),(135,143),(136,142),(137,141),(138,140)], [(1,40,102,126,19,58,84,144),(2,41,103,127,20,59,85,109),(3,42,104,128,21,60,86,110),(4,43,105,129,22,61,87,111),(5,44,106,130,23,62,88,112),(6,45,107,131,24,63,89,113),(7,46,108,132,25,64,90,114),(8,47,73,133,26,65,91,115),(9,48,74,134,27,66,92,116),(10,49,75,135,28,67,93,117),(11,50,76,136,29,68,94,118),(12,51,77,137,30,69,95,119),(13,52,78,138,31,70,96,120),(14,53,79,139,32,71,97,121),(15,54,80,140,33,72,98,122),(16,55,81,141,34,37,99,123),(17,56,82,142,35,38,100,124),(18,57,83,143,36,39,101,125)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | D9 | C4×S3 | C4×S3 | C8○D4 | D18 | D18 | C4×D9 | C4×D9 | C8○D12 | D36.2C4 |
| kernel | D36.2C4 | C8×D9 | C8⋊D9 | C4.Dic9 | C2×C72 | D36⋊5C2 | Dic18 | D36 | C9⋊D4 | C2×C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C9 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 4 | 6 | 3 | 6 | 6 | 8 | 24 |
Matrix representation of D36.2C4 ►in GL4(𝔽73) generated by
| 66 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 31 | 45 |
| 0 | 0 | 28 | 3 |
| 1 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
| 51 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,31,28,0,0,45,3],[1,0,0,0,1,72,0,0,0,0,72,0,0,0,1,1],[51,0,0,0,0,51,0,0,0,0,72,0,0,0,0,72] >;
D36.2C4 in GAP, Magma, Sage, TeX
D_{36}._2C_4 % in TeX
G:=Group("D36.2C4"); // GroupNames label
G:=SmallGroup(288,112);
// by ID
G=gap.SmallGroup(288,112);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^2=1,c^4=a^18,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations