metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic9.6D4, C22⋊2Dic18, C23.17D18, (C2×C18)⋊Q8, C2.6(D4×D9), C4⋊Dic9⋊2C2, (C2×C4).5D18, (C2×C12).1D6, C6.76(S3×D4), C9⋊1(C22⋊Q8), C18.4(C2×Q8), Dic9⋊C4⋊4C2, C18.16(C2×D4), C22⋊C4.1D9, (C2×C6).3Dic6, (C2×Dic18)⋊2C2, (C2×C36).1C22, C2.6(C2×Dic18), C6.31(C2×Dic6), (C22×C6).38D6, C18.21(C4○D4), C2.6(D4⋊2D9), (C2×C18).19C23, C6.73(D4⋊2S3), C18.D4.2C2, C3.(Dic3.D4), (C22×C18).8C22, (C22×Dic9).3C2, C22.39(C22×D9), (C2×Dic9).24C22, (C3×C22⋊C4).2S3, (C9×C22⋊C4).1C2, (C2×C6).176(C22×S3), SmallGroup(288,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊2Dic18
G = < a,b,c,d | a2=b2=c36=1, d2=c18, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 420 in 111 conjugacy classes, 46 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C9, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C18, C18, Dic6, C2×Dic3, C2×C12, C22×C6, C22⋊Q8, Dic9, Dic9, C36, C2×C18, C2×C18, C2×C18, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, Dic18, C2×Dic9, C2×Dic9, C2×C36, C22×C18, Dic3.D4, Dic9⋊C4, C4⋊Dic9, C18.D4, C9×C22⋊C4, C2×Dic18, C22×Dic9, C22⋊2Dic18
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D9, Dic6, C22×S3, C22⋊Q8, D18, C2×Dic6, S3×D4, D4⋊2S3, Dic18, C22×D9, Dic3.D4, C2×Dic18, D4×D9, D4⋊2D9, C22⋊2Dic18
►On 144 points
Generators in S144
(1 19)(2 73)(3 21)(4 75)(5 23)(6 77)(7 25)(8 79)(9 27)(10 81)(11 29)(12 83)(13 31)(14 85)(15 33)(16 87)(17 35)(18 89)(20 91)(22 93)(24 95)(26 97)(28 99)(30 101)(32 103)(34 105)(36 107)(37 128)(38 56)(39 130)(40 58)(41 132)(42 60)(43 134)(44 62)(45 136)(46 64)(47 138)(48 66)(49 140)(50 68)(51 142)(52 70)(53 144)(54 72)(55 110)(57 112)(59 114)(61 116)(63 118)(65 120)(67 122)(69 124)(71 126)(74 92)(76 94)(78 96)(80 98)(82 100)(84 102)(86 104)(88 106)(90 108)(109 127)(111 129)(113 131)(115 133)(117 135)(119 137)(121 139)(123 141)(125 143)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 109)
(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)
(1 133 19 115)(2 132 20 114)(3 131 21 113)(4 130 22 112)(5 129 23 111)(6 128 24 110)(7 127 25 109)(8 126 26 144)(9 125 27 143)(10 124 28 142)(11 123 29 141)(12 122 30 140)(13 121 31 139)(14 120 32 138)(15 119 33 137)(16 118 34 136)(17 117 35 135)(18 116 36 134)(37 95 55 77)(38 94 56 76)(39 93 57 75)(40 92 58 74)(41 91 59 73)(42 90 60 108)(43 89 61 107)(44 88 62 106)(45 87 63 105)(46 86 64 104)(47 85 65 103)(48 84 66 102)(49 83 67 101)(50 82 68 100)(51 81 69 99)(52 80 70 98)(53 79 71 97)(54 78 72 96)
G:=sub<Sym(144)| (1,19)(2,73)(3,21)(4,75)(5,23)(6,77)(7,25)(8,79)(9,27)(10,81)(11,29)(12,83)(13,31)(14,85)(15,33)(16,87)(17,35)(18,89)(20,91)(22,93)(24,95)(26,97)(28,99)(30,101)(32,103)(34,105)(36,107)(37,128)(38,56)(39,130)(40,58)(41,132)(42,60)(43,134)(44,62)(45,136)(46,64)(47,138)(48,66)(49,140)(50,68)(51,142)(52,70)(53,144)(54,72)(55,110)(57,112)(59,114)(61,116)(63,118)(65,120)(67,122)(69,124)(71,126)(74,92)(76,94)(78,96)(80,98)(82,100)(84,102)(86,104)(88,106)(90,108)(109,127)(111,129)(113,131)(115,133)(117,135)(119,137)(121,139)(123,141)(125,143), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,109), (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), (1,133,19,115)(2,132,20,114)(3,131,21,113)(4,130,22,112)(5,129,23,111)(6,128,24,110)(7,127,25,109)(8,126,26,144)(9,125,27,143)(10,124,28,142)(11,123,29,141)(12,122,30,140)(13,121,31,139)(14,120,32,138)(15,119,33,137)(16,118,34,136)(17,117,35,135)(18,116,36,134)(37,95,55,77)(38,94,56,76)(39,93,57,75)(40,92,58,74)(41,91,59,73)(42,90,60,108)(43,89,61,107)(44,88,62,106)(45,87,63,105)(46,86,64,104)(47,85,65,103)(48,84,66,102)(49,83,67,101)(50,82,68,100)(51,81,69,99)(52,80,70,98)(53,79,71,97)(54,78,72,96)>;
G:=Group( (1,19)(2,73)(3,21)(4,75)(5,23)(6,77)(7,25)(8,79)(9,27)(10,81)(11,29)(12,83)(13,31)(14,85)(15,33)(16,87)(17,35)(18,89)(20,91)(22,93)(24,95)(26,97)(28,99)(30,101)(32,103)(34,105)(36,107)(37,128)(38,56)(39,130)(40,58)(41,132)(42,60)(43,134)(44,62)(45,136)(46,64)(47,138)(48,66)(49,140)(50,68)(51,142)(52,70)(53,144)(54,72)(55,110)(57,112)(59,114)(61,116)(63,118)(65,120)(67,122)(69,124)(71,126)(74,92)(76,94)(78,96)(80,98)(82,100)(84,102)(86,104)(88,106)(90,108)(109,127)(111,129)(113,131)(115,133)(117,135)(119,137)(121,139)(123,141)(125,143), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,109), (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), (1,133,19,115)(2,132,20,114)(3,131,21,113)(4,130,22,112)(5,129,23,111)(6,128,24,110)(7,127,25,109)(8,126,26,144)(9,125,27,143)(10,124,28,142)(11,123,29,141)(12,122,30,140)(13,121,31,139)(14,120,32,138)(15,119,33,137)(16,118,34,136)(17,117,35,135)(18,116,36,134)(37,95,55,77)(38,94,56,76)(39,93,57,75)(40,92,58,74)(41,91,59,73)(42,90,60,108)(43,89,61,107)(44,88,62,106)(45,87,63,105)(46,86,64,104)(47,85,65,103)(48,84,66,102)(49,83,67,101)(50,82,68,100)(51,81,69,99)(52,80,70,98)(53,79,71,97)(54,78,72,96) );
G=PermutationGroup([[(1,19),(2,73),(3,21),(4,75),(5,23),(6,77),(7,25),(8,79),(9,27),(10,81),(11,29),(12,83),(13,31),(14,85),(15,33),(16,87),(17,35),(18,89),(20,91),(22,93),(24,95),(26,97),(28,99),(30,101),(32,103),(34,105),(36,107),(37,128),(38,56),(39,130),(40,58),(41,132),(42,60),(43,134),(44,62),(45,136),(46,64),(47,138),(48,66),(49,140),(50,68),(51,142),(52,70),(53,144),(54,72),(55,110),(57,112),(59,114),(61,116),(63,118),(65,120),(67,122),(69,124),(71,126),(74,92),(76,94),(78,96),(80,98),(82,100),(84,102),(86,104),(88,106),(90,108),(109,127),(111,129),(113,131),(115,133),(117,135),(119,137),(121,139),(123,141),(125,143)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,110),(38,111),(39,112),(40,113),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,141),(69,142),(70,143),(71,144),(72,109)], [(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)], [(1,133,19,115),(2,132,20,114),(3,131,21,113),(4,130,22,112),(5,129,23,111),(6,128,24,110),(7,127,25,109),(8,126,26,144),(9,125,27,143),(10,124,28,142),(11,123,29,141),(12,122,30,140),(13,121,31,139),(14,120,32,138),(15,119,33,137),(16,118,34,136),(17,117,35,135),(18,116,36,134),(37,95,55,77),(38,94,56,76),(39,93,57,75),(40,92,58,74),(41,91,59,73),(42,90,60,108),(43,89,61,107),(44,88,62,106),(45,87,63,105),(46,86,64,104),(47,85,65,103),(48,84,66,102),(49,83,67,101),(50,82,68,100),(51,81,69,99),(52,80,70,98),(53,79,71,97),(54,78,72,96)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | D9 | Dic6 | D18 | D18 | Dic18 | S3×D4 | D4⋊2S3 | D4×D9 | D4⋊2D9 |
| kernel | C22⋊2Dic18 | Dic9⋊C4 | C4⋊Dic9 | C18.D4 | C9×C22⋊C4 | C2×Dic18 | C22×Dic9 | C3×C22⋊C4 | Dic9 | C2×C18 | C2×C12 | C22×C6 | C18 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 3 | 4 | 6 | 3 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of C22⋊2Dic18 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 20 |
| 0 | 0 | 0 | 0 | 17 | 31 |
| 31 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 |
| 0 | 0 | 0 | 0 | 17 | 26 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,36,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,1,0,0,0,0,0,0,0,11,17,0,0,0,0,20,31],[31,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,31,0,0,0,0,0,0,11,17,0,0,0,0,6,26] >;
C22⋊2Dic18 in GAP, Magma, Sage, TeX
C_2^2\rtimes_2{\rm Dic}_{18} % in TeX
G:=Group("C2^2:2Dic18"); // GroupNames label
G:=SmallGroup(288,88);
// by ID
G=gap.SmallGroup(288,88);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,254,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^36=1,d^2=c^18,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations