metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.8D18, C4⋊Dic9⋊3C2, (C2×C4).6D18, Dic9⋊C4⋊8C2, C22⋊C4.2D9, C9⋊2(C42⋊2C2), (C4×Dic9)⋊10C2, C18.7(C4○D4), (C2×C12).175D6, (C2×C36).2C22, (C22×C6).39D6, C6.77(C4○D12), C2.7(D4⋊2D9), (C2×C18).20C23, C3.(C23.8D6), C6.74(D4⋊2S3), C2.9(D36⋊5C2), C18.D4.3C2, (C22×C18).9C22, C22.40(C22×D9), (C2×Dic9).25C22, (C9×C22⋊C4).2C2, (C3×C22⋊C4).6S3, (C2×C6).177(C22×S3), SmallGroup(288,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.8D18
G = < a,b,c,d,e | a2=b2=c2=1, d18=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d17 >
Subgroups: 316 in 90 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C18, C18, C2×Dic3, C2×C12, C22×C6, C42⋊2C2, Dic9, C36, C2×C18, C2×C18, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic9, C2×C36, C22×C18, C23.8D6, C4×Dic9, Dic9⋊C4, C4⋊Dic9, C18.D4, C9×C22⋊C4, C23.8D18
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, C42⋊2C2, D18, C4○D12, D4⋊2S3, C22×D9, C23.8D6, D36⋊5C2, D4⋊2D9, C23.8D18
►On 144 points
Generators in S144
(2 128)(4 130)(6 132)(8 134)(10 136)(12 138)(14 140)(16 142)(18 144)(20 110)(22 112)(24 114)(26 116)(28 118)(30 120)(32 122)(34 124)(36 126)(37 102)(38 56)(39 104)(40 58)(41 106)(42 60)(43 108)(44 62)(45 74)(46 64)(47 76)(48 66)(49 78)(50 68)(51 80)(52 70)(53 82)(54 72)(55 84)(57 86)(59 88)(61 90)(63 92)(65 94)(67 96)(69 98)(71 100)(73 91)(75 93)(77 95)(79 97)(81 99)(83 101)(85 103)(87 105)(89 107)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)(57 104)(58 105)(59 106)(60 107)(61 108)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)
(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 62 109 91)(2 43 110 108)(3 60 111 89)(4 41 112 106)(5 58 113 87)(6 39 114 104)(7 56 115 85)(8 37 116 102)(9 54 117 83)(10 71 118 100)(11 52 119 81)(12 69 120 98)(13 50 121 79)(14 67 122 96)(15 48 123 77)(16 65 124 94)(17 46 125 75)(18 63 126 92)(19 44 127 73)(20 61 128 90)(21 42 129 107)(22 59 130 88)(23 40 131 105)(24 57 132 86)(25 38 133 103)(26 55 134 84)(27 72 135 101)(28 53 136 82)(29 70 137 99)(30 51 138 80)(31 68 139 97)(32 49 140 78)(33 66 141 95)(34 47 142 76)(35 64 143 93)(36 45 144 74)
G:=sub<Sym(144)| (2,128)(4,130)(6,132)(8,134)(10,136)(12,138)(14,140)(16,142)(18,144)(20,110)(22,112)(24,114)(26,116)(28,118)(30,120)(32,122)(34,124)(36,126)(37,102)(38,56)(39,104)(40,58)(41,106)(42,60)(43,108)(44,62)(45,74)(46,64)(47,76)(48,66)(49,78)(50,68)(51,80)(52,70)(53,82)(54,72)(55,84)(57,86)(59,88)(61,90)(63,92)(65,94)(67,96)(69,98)(71,100)(73,91)(75,93)(77,95)(79,97)(81,99)(83,101)(85,103)(87,105)(89,107), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,108)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83), (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), (1,62,109,91)(2,43,110,108)(3,60,111,89)(4,41,112,106)(5,58,113,87)(6,39,114,104)(7,56,115,85)(8,37,116,102)(9,54,117,83)(10,71,118,100)(11,52,119,81)(12,69,120,98)(13,50,121,79)(14,67,122,96)(15,48,123,77)(16,65,124,94)(17,46,125,75)(18,63,126,92)(19,44,127,73)(20,61,128,90)(21,42,129,107)(22,59,130,88)(23,40,131,105)(24,57,132,86)(25,38,133,103)(26,55,134,84)(27,72,135,101)(28,53,136,82)(29,70,137,99)(30,51,138,80)(31,68,139,97)(32,49,140,78)(33,66,141,95)(34,47,142,76)(35,64,143,93)(36,45,144,74)>;
G:=Group( (2,128)(4,130)(6,132)(8,134)(10,136)(12,138)(14,140)(16,142)(18,144)(20,110)(22,112)(24,114)(26,116)(28,118)(30,120)(32,122)(34,124)(36,126)(37,102)(38,56)(39,104)(40,58)(41,106)(42,60)(43,108)(44,62)(45,74)(46,64)(47,76)(48,66)(49,78)(50,68)(51,80)(52,70)(53,82)(54,72)(55,84)(57,86)(59,88)(61,90)(63,92)(65,94)(67,96)(69,98)(71,100)(73,91)(75,93)(77,95)(79,97)(81,99)(83,101)(85,103)(87,105)(89,107), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,108)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83), (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), (1,62,109,91)(2,43,110,108)(3,60,111,89)(4,41,112,106)(5,58,113,87)(6,39,114,104)(7,56,115,85)(8,37,116,102)(9,54,117,83)(10,71,118,100)(11,52,119,81)(12,69,120,98)(13,50,121,79)(14,67,122,96)(15,48,123,77)(16,65,124,94)(17,46,125,75)(18,63,126,92)(19,44,127,73)(20,61,128,90)(21,42,129,107)(22,59,130,88)(23,40,131,105)(24,57,132,86)(25,38,133,103)(26,55,134,84)(27,72,135,101)(28,53,136,82)(29,70,137,99)(30,51,138,80)(31,68,139,97)(32,49,140,78)(33,66,141,95)(34,47,142,76)(35,64,143,93)(36,45,144,74) );
G=PermutationGroup([[(2,128),(4,130),(6,132),(8,134),(10,136),(12,138),(14,140),(16,142),(18,144),(20,110),(22,112),(24,114),(26,116),(28,118),(30,120),(32,122),(34,124),(36,126),(37,102),(38,56),(39,104),(40,58),(41,106),(42,60),(43,108),(44,62),(45,74),(46,64),(47,76),(48,66),(49,78),(50,68),(51,80),(52,70),(53,82),(54,72),(55,84),(57,86),(59,88),(61,90),(63,92),(65,94),(67,96),(69,98),(71,100),(73,91),(75,93),(77,95),(79,97),(81,99),(83,101),(85,103),(87,105),(89,107)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138),(121,139),(122,140),(123,141),(124,142),(125,143),(126,144)], [(1,127),(2,128),(3,129),(4,130),(5,131),(6,132),(7,133),(8,134),(9,135),(10,136),(11,137),(12,138),(13,139),(14,140),(15,141),(16,142),(17,143),(18,144),(19,109),(20,110),(21,111),(22,112),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,121),(32,122),(33,123),(34,124),(35,125),(36,126),(37,84),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103),(57,104),(58,105),(59,106),(60,107),(61,108),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83)], [(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)], [(1,62,109,91),(2,43,110,108),(3,60,111,89),(4,41,112,106),(5,58,113,87),(6,39,114,104),(7,56,115,85),(8,37,116,102),(9,54,117,83),(10,71,118,100),(11,52,119,81),(12,69,120,98),(13,50,121,79),(14,67,122,96),(15,48,123,77),(16,65,124,94),(17,46,125,75),(18,63,126,92),(19,44,127,73),(20,61,128,90),(21,42,129,107),(22,59,130,88),(23,40,131,105),(24,57,132,86),(25,38,133,103),(26,55,134,84),(27,72,135,101),(28,53,136,82),(29,70,137,99),(30,51,138,80),(31,68,139,97),(32,49,140,78),(33,66,141,95),(34,47,142,76),(35,64,143,93),(36,45,144,74)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 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 | 4 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | D9 | D18 | D18 | C4○D12 | D36⋊5C2 | D4⋊2S3 | D4⋊2D9 |
| kernel | C23.8D18 | C4×Dic9 | Dic9⋊C4 | C4⋊Dic9 | C18.D4 | C9×C22⋊C4 | C3×C22⋊C4 | C2×C12 | C22×C6 | C18 | C22⋊C4 | C2×C4 | C23 | C6 | C2 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 6 | 3 | 6 | 3 | 4 | 12 | 2 | 6 |
Matrix representation of C23.8D18 ►in GL4(𝔽37) generated by
| 1 | 4 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 13 | 29 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 31 | 28 |
| 0 | 0 | 0 | 6 |
| 12 | 18 | 0 | 0 |
| 31 | 25 | 0 | 0 |
| 0 | 0 | 36 | 17 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [1,0,0,0,4,36,0,0,0,0,1,11,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[13,0,0,0,29,17,0,0,0,0,31,0,0,0,28,6],[12,31,0,0,18,25,0,0,0,0,36,0,0,0,17,1] >;
C23.8D18 in GAP, Magma, Sage, TeX
C_2^3._8D_{18} % in TeX
G:=Group("C2^3.8D18"); // GroupNames label
G:=SmallGroup(288,89);
// by ID
G=gap.SmallGroup(288,89);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^18=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^17>;
// generators/relations