metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.10D18, Q8.16D18, C9⋊22- 1+4, C36.27C23, C18.13C24, D18.8C23, D36.14C22, Dic9.8C23, Dic18.14C22, C4○D4⋊6D9, (Q8×D9)⋊5C2, C3.(Q8○D12), C9⋊D4.C22, D4⋊2D9⋊5C2, (C3×D4).40D6, (C2×C4).23D18, (C3×Q8).64D6, D36⋊5C2⋊9C2, (C2×C12).106D6, C6.50(S3×C23), (C2×C18).5C23, (C4×D9).6C22, C2.14(C23×D9), C4.34(C22×D9), (C2×Dic18)⋊14C2, (C2×C36).52C22, (D4×C9).10C22, (Q8×C9).11C22, C22.4(C22×D9), C12.188(C22×S3), (C2×Dic9).19C22, (C9×C4○D4)⋊5C2, (C3×C4○D4).17S3, (C2×C6).11(C22×S3), SmallGroup(288,364)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.10D18
G = < a,b,c,d | a4=b2=1, c18=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c17 >
Subgroups: 784 in 219 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C6, C2×Q8, C4○D4, C4○D4, D9, C18, C18, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, 2- 1+4, Dic9, C36, C36, D18, C2×C18, C2×Dic6, C4○D12, D4⋊2S3, S3×Q8, C3×C4○D4, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, D4×C9, Q8×C9, Q8○D12, C2×Dic18, D36⋊5C2, D4⋊2D9, Q8×D9, C9×C4○D4, D4.10D18
Quotients: C1, C2, C22, S3, C23, D6, C24, D9, C22×S3, 2- 1+4, D18, S3×C23, C22×D9, Q8○D12, C23×D9, D4.10D18
►On 144 points
Generators in S144
(1 76 19 94)(2 95 20 77)(3 78 21 96)(4 97 22 79)(5 80 23 98)(6 99 24 81)(7 82 25 100)(8 101 26 83)(9 84 27 102)(10 103 28 85)(11 86 29 104)(12 105 30 87)(13 88 31 106)(14 107 32 89)(15 90 33 108)(16 73 34 91)(17 92 35 74)(18 75 36 93)(37 111 55 129)(38 130 56 112)(39 113 57 131)(40 132 58 114)(41 115 59 133)(42 134 60 116)(43 117 61 135)(44 136 62 118)(45 119 63 137)(46 138 64 120)(47 121 65 139)(48 140 66 122)(49 123 67 141)(50 142 68 124)(51 125 69 143)(52 144 70 126)(53 127 71 109)(54 110 72 128)
(1 112)(2 131)(3 114)(4 133)(5 116)(6 135)(7 118)(8 137)(9 120)(10 139)(11 122)(12 141)(13 124)(14 143)(15 126)(16 109)(17 128)(18 111)(19 130)(20 113)(21 132)(22 115)(23 134)(24 117)(25 136)(26 119)(27 138)(28 121)(29 140)(30 123)(31 142)(32 125)(33 144)(34 127)(35 110)(36 129)(37 75)(38 94)(39 77)(40 96)(41 79)(42 98)(43 81)(44 100)(45 83)(46 102)(47 85)(48 104)(49 87)(50 106)(51 89)(52 108)(53 91)(54 74)(55 93)(56 76)(57 95)(58 78)(59 97)(60 80)(61 99)(62 82)(63 101)(64 84)(65 103)(66 86)(67 105)(68 88)(69 107)(70 90)(71 73)(72 92)
(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 18 19 36)(2 35 20 17)(3 16 21 34)(4 33 22 15)(5 14 23 32)(6 31 24 13)(7 12 25 30)(8 29 26 11)(9 10 27 28)(37 56 55 38)(39 54 57 72)(40 71 58 53)(41 52 59 70)(42 69 60 51)(43 50 61 68)(44 67 62 49)(45 48 63 66)(46 65 64 47)(73 78 91 96)(74 95 92 77)(75 76 93 94)(79 108 97 90)(80 89 98 107)(81 106 99 88)(82 87 100 105)(83 104 101 86)(84 85 102 103)(109 132 127 114)(110 113 128 131)(111 130 129 112)(115 126 133 144)(116 143 134 125)(117 124 135 142)(118 141 136 123)(119 122 137 140)(120 139 138 121)
G:=sub<Sym(144)| (1,76,19,94)(2,95,20,77)(3,78,21,96)(4,97,22,79)(5,80,23,98)(6,99,24,81)(7,82,25,100)(8,101,26,83)(9,84,27,102)(10,103,28,85)(11,86,29,104)(12,105,30,87)(13,88,31,106)(14,107,32,89)(15,90,33,108)(16,73,34,91)(17,92,35,74)(18,75,36,93)(37,111,55,129)(38,130,56,112)(39,113,57,131)(40,132,58,114)(41,115,59,133)(42,134,60,116)(43,117,61,135)(44,136,62,118)(45,119,63,137)(46,138,64,120)(47,121,65,139)(48,140,66,122)(49,123,67,141)(50,142,68,124)(51,125,69,143)(52,144,70,126)(53,127,71,109)(54,110,72,128), (1,112)(2,131)(3,114)(4,133)(5,116)(6,135)(7,118)(8,137)(9,120)(10,139)(11,122)(12,141)(13,124)(14,143)(15,126)(16,109)(17,128)(18,111)(19,130)(20,113)(21,132)(22,115)(23,134)(24,117)(25,136)(26,119)(27,138)(28,121)(29,140)(30,123)(31,142)(32,125)(33,144)(34,127)(35,110)(36,129)(37,75)(38,94)(39,77)(40,96)(41,79)(42,98)(43,81)(44,100)(45,83)(46,102)(47,85)(48,104)(49,87)(50,106)(51,89)(52,108)(53,91)(54,74)(55,93)(56,76)(57,95)(58,78)(59,97)(60,80)(61,99)(62,82)(63,101)(64,84)(65,103)(66,86)(67,105)(68,88)(69,107)(70,90)(71,73)(72,92), (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), (1,18,19,36)(2,35,20,17)(3,16,21,34)(4,33,22,15)(5,14,23,32)(6,31,24,13)(7,12,25,30)(8,29,26,11)(9,10,27,28)(37,56,55,38)(39,54,57,72)(40,71,58,53)(41,52,59,70)(42,69,60,51)(43,50,61,68)(44,67,62,49)(45,48,63,66)(46,65,64,47)(73,78,91,96)(74,95,92,77)(75,76,93,94)(79,108,97,90)(80,89,98,107)(81,106,99,88)(82,87,100,105)(83,104,101,86)(84,85,102,103)(109,132,127,114)(110,113,128,131)(111,130,129,112)(115,126,133,144)(116,143,134,125)(117,124,135,142)(118,141,136,123)(119,122,137,140)(120,139,138,121)>;
G:=Group( (1,76,19,94)(2,95,20,77)(3,78,21,96)(4,97,22,79)(5,80,23,98)(6,99,24,81)(7,82,25,100)(8,101,26,83)(9,84,27,102)(10,103,28,85)(11,86,29,104)(12,105,30,87)(13,88,31,106)(14,107,32,89)(15,90,33,108)(16,73,34,91)(17,92,35,74)(18,75,36,93)(37,111,55,129)(38,130,56,112)(39,113,57,131)(40,132,58,114)(41,115,59,133)(42,134,60,116)(43,117,61,135)(44,136,62,118)(45,119,63,137)(46,138,64,120)(47,121,65,139)(48,140,66,122)(49,123,67,141)(50,142,68,124)(51,125,69,143)(52,144,70,126)(53,127,71,109)(54,110,72,128), (1,112)(2,131)(3,114)(4,133)(5,116)(6,135)(7,118)(8,137)(9,120)(10,139)(11,122)(12,141)(13,124)(14,143)(15,126)(16,109)(17,128)(18,111)(19,130)(20,113)(21,132)(22,115)(23,134)(24,117)(25,136)(26,119)(27,138)(28,121)(29,140)(30,123)(31,142)(32,125)(33,144)(34,127)(35,110)(36,129)(37,75)(38,94)(39,77)(40,96)(41,79)(42,98)(43,81)(44,100)(45,83)(46,102)(47,85)(48,104)(49,87)(50,106)(51,89)(52,108)(53,91)(54,74)(55,93)(56,76)(57,95)(58,78)(59,97)(60,80)(61,99)(62,82)(63,101)(64,84)(65,103)(66,86)(67,105)(68,88)(69,107)(70,90)(71,73)(72,92), (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), (1,18,19,36)(2,35,20,17)(3,16,21,34)(4,33,22,15)(5,14,23,32)(6,31,24,13)(7,12,25,30)(8,29,26,11)(9,10,27,28)(37,56,55,38)(39,54,57,72)(40,71,58,53)(41,52,59,70)(42,69,60,51)(43,50,61,68)(44,67,62,49)(45,48,63,66)(46,65,64,47)(73,78,91,96)(74,95,92,77)(75,76,93,94)(79,108,97,90)(80,89,98,107)(81,106,99,88)(82,87,100,105)(83,104,101,86)(84,85,102,103)(109,132,127,114)(110,113,128,131)(111,130,129,112)(115,126,133,144)(116,143,134,125)(117,124,135,142)(118,141,136,123)(119,122,137,140)(120,139,138,121) );
G=PermutationGroup([[(1,76,19,94),(2,95,20,77),(3,78,21,96),(4,97,22,79),(5,80,23,98),(6,99,24,81),(7,82,25,100),(8,101,26,83),(9,84,27,102),(10,103,28,85),(11,86,29,104),(12,105,30,87),(13,88,31,106),(14,107,32,89),(15,90,33,108),(16,73,34,91),(17,92,35,74),(18,75,36,93),(37,111,55,129),(38,130,56,112),(39,113,57,131),(40,132,58,114),(41,115,59,133),(42,134,60,116),(43,117,61,135),(44,136,62,118),(45,119,63,137),(46,138,64,120),(47,121,65,139),(48,140,66,122),(49,123,67,141),(50,142,68,124),(51,125,69,143),(52,144,70,126),(53,127,71,109),(54,110,72,128)], [(1,112),(2,131),(3,114),(4,133),(5,116),(6,135),(7,118),(8,137),(9,120),(10,139),(11,122),(12,141),(13,124),(14,143),(15,126),(16,109),(17,128),(18,111),(19,130),(20,113),(21,132),(22,115),(23,134),(24,117),(25,136),(26,119),(27,138),(28,121),(29,140),(30,123),(31,142),(32,125),(33,144),(34,127),(35,110),(36,129),(37,75),(38,94),(39,77),(40,96),(41,79),(42,98),(43,81),(44,100),(45,83),(46,102),(47,85),(48,104),(49,87),(50,106),(51,89),(52,108),(53,91),(54,74),(55,93),(56,76),(57,95),(58,78),(59,97),(60,80),(61,99),(62,82),(63,101),(64,84),(65,103),(66,86),(67,105),(68,88),(69,107),(70,90),(71,73),(72,92)], [(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)], [(1,18,19,36),(2,35,20,17),(3,16,21,34),(4,33,22,15),(5,14,23,32),(6,31,24,13),(7,12,25,30),(8,29,26,11),(9,10,27,28),(37,56,55,38),(39,54,57,72),(40,71,58,53),(41,52,59,70),(42,69,60,51),(43,50,61,68),(44,67,62,49),(45,48,63,66),(46,65,64,47),(73,78,91,96),(74,95,92,77),(75,76,93,94),(79,108,97,90),(80,89,98,107),(81,106,99,88),(82,87,100,105),(83,104,101,86),(84,85,102,103),(109,132,127,114),(110,113,128,131),(111,130,129,112),(115,126,133,144),(116,143,134,125),(117,124,135,142),(118,141,136,123),(119,122,137,140),(120,139,138,121)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D9 | D18 | D18 | D18 | 2- 1+4 | Q8○D12 | D4.10D18 |
| kernel | D4.10D18 | C2×Dic18 | D36⋊5C2 | D4⋊2D9 | Q8×D9 | C9×C4○D4 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C9 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 1 | 3 | 3 | 1 | 3 | 9 | 9 | 3 | 1 | 2 | 6 |
Matrix representation of D4.10D18 ►in GL4(𝔽37) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 36 | 2 | 14 | 9 |
| 35 | 1 | 28 | 23 |
| 14 | 9 | 1 | 35 |
| 28 | 23 | 2 | 36 |
| 19 | 2 | 4 | 16 |
| 35 | 21 | 21 | 20 |
| 4 | 16 | 18 | 35 |
| 21 | 20 | 2 | 16 |
| 35 | 18 | 21 | 33 |
| 16 | 2 | 17 | 16 |
| 21 | 33 | 2 | 19 |
| 17 | 16 | 21 | 35 |
G:=sub<GL(4,GF(37))| [0,0,36,0,0,0,0,36,1,0,0,0,0,1,0,0],[36,35,14,28,2,1,9,23,14,28,1,2,9,23,35,36],[19,35,4,21,2,21,16,20,4,21,18,2,16,20,35,16],[35,16,21,17,18,2,33,16,21,17,2,21,33,16,19,35] >;
D4.10D18 in GAP, Magma, Sage, TeX
D_4._{10}D_{18} % in TeX
G:=Group("D4.10D18"); // GroupNames label
G:=SmallGroup(288,364);
// by ID
G=gap.SmallGroup(288,364);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^18=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^17>;
// generators/relations