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 [×3], C2, C3, C4 [×6], C22, C22 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C23, C9, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], C18 [×3], C18, C2×Dic3 [×4], C2×C12 [×2], C22×C6, C42⋊2C2, Dic9 [×4], C36 [×2], C2×C18, C2×C18 [×3], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, C2×Dic9 [×4], C2×C36 [×2], C22×C18, C23.8D6, C4×Dic9, Dic9⋊C4 [×2], C4⋊Dic9, C18.D4 [×2], C9×C22⋊C4, C23.8D18
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4 [×3], D9, C22×S3, C42⋊2C2, D18 [×3], C4○D12, D4⋊2S3 [×2], C22×D9, C23.8D6, D36⋊5C2, D4⋊2D9 [×2], C23.8D18
►On 144 points
Generators in S144
(2 44)(4 46)(6 48)(8 50)(10 52)(12 54)(14 56)(16 58)(18 60)(20 62)(22 64)(24 66)(26 68)(28 70)(30 72)(32 38)(34 40)(36 42)(73 91)(74 116)(75 93)(76 118)(77 95)(78 120)(79 97)(80 122)(81 99)(82 124)(83 101)(84 126)(85 103)(86 128)(87 105)(88 130)(89 107)(90 132)(92 134)(94 136)(96 138)(98 140)(100 142)(102 144)(104 110)(106 112)(108 114)(109 127)(111 129)(113 131)(115 133)(117 135)(119 137)(121 139)(123 141)(125 143)
(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 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)(82 142)(83 143)(84 144)(85 109)(86 110)(87 111)(88 112)(89 113)(90 114)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(97 121)(98 122)(99 123)(100 124)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)
(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 99 61 141)(2 80 62 122)(3 97 63 139)(4 78 64 120)(5 95 65 137)(6 76 66 118)(7 93 67 135)(8 74 68 116)(9 91 69 133)(10 108 70 114)(11 89 71 131)(12 106 72 112)(13 87 37 129)(14 104 38 110)(15 85 39 127)(16 102 40 144)(17 83 41 125)(18 100 42 142)(19 81 43 123)(20 98 44 140)(21 79 45 121)(22 96 46 138)(23 77 47 119)(24 94 48 136)(25 75 49 117)(26 92 50 134)(27 73 51 115)(28 90 52 132)(29 107 53 113)(30 88 54 130)(31 105 55 111)(32 86 56 128)(33 103 57 109)(34 84 58 126)(35 101 59 143)(36 82 60 124)
G:=sub<Sym(144)| (2,44)(4,46)(6,48)(8,50)(10,52)(12,54)(14,56)(16,58)(18,60)(20,62)(22,64)(24,66)(26,68)(28,70)(30,72)(32,38)(34,40)(36,42)(73,91)(74,116)(75,93)(76,118)(77,95)(78,120)(79,97)(80,122)(81,99)(82,124)(83,101)(84,126)(85,103)(86,128)(87,105)(88,130)(89,107)(90,132)(92,134)(94,136)(96,138)(98,140)(100,142)(102,144)(104,110)(106,112)(108,114)(109,127)(111,129)(113,131)(115,133)(117,135)(119,137)(121,139)(123,141)(125,143), (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), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(83,143)(84,144)(85,109)(86,110)(87,111)(88,112)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(97,121)(98,122)(99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,131)(108,132), (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), (1,99,61,141)(2,80,62,122)(3,97,63,139)(4,78,64,120)(5,95,65,137)(6,76,66,118)(7,93,67,135)(8,74,68,116)(9,91,69,133)(10,108,70,114)(11,89,71,131)(12,106,72,112)(13,87,37,129)(14,104,38,110)(15,85,39,127)(16,102,40,144)(17,83,41,125)(18,100,42,142)(19,81,43,123)(20,98,44,140)(21,79,45,121)(22,96,46,138)(23,77,47,119)(24,94,48,136)(25,75,49,117)(26,92,50,134)(27,73,51,115)(28,90,52,132)(29,107,53,113)(30,88,54,130)(31,105,55,111)(32,86,56,128)(33,103,57,109)(34,84,58,126)(35,101,59,143)(36,82,60,124)>;
G:=Group( (2,44)(4,46)(6,48)(8,50)(10,52)(12,54)(14,56)(16,58)(18,60)(20,62)(22,64)(24,66)(26,68)(28,70)(30,72)(32,38)(34,40)(36,42)(73,91)(74,116)(75,93)(76,118)(77,95)(78,120)(79,97)(80,122)(81,99)(82,124)(83,101)(84,126)(85,103)(86,128)(87,105)(88,130)(89,107)(90,132)(92,134)(94,136)(96,138)(98,140)(100,142)(102,144)(104,110)(106,112)(108,114)(109,127)(111,129)(113,131)(115,133)(117,135)(119,137)(121,139)(123,141)(125,143), (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), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(83,143)(84,144)(85,109)(86,110)(87,111)(88,112)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(97,121)(98,122)(99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,131)(108,132), (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), (1,99,61,141)(2,80,62,122)(3,97,63,139)(4,78,64,120)(5,95,65,137)(6,76,66,118)(7,93,67,135)(8,74,68,116)(9,91,69,133)(10,108,70,114)(11,89,71,131)(12,106,72,112)(13,87,37,129)(14,104,38,110)(15,85,39,127)(16,102,40,144)(17,83,41,125)(18,100,42,142)(19,81,43,123)(20,98,44,140)(21,79,45,121)(22,96,46,138)(23,77,47,119)(24,94,48,136)(25,75,49,117)(26,92,50,134)(27,73,51,115)(28,90,52,132)(29,107,53,113)(30,88,54,130)(31,105,55,111)(32,86,56,128)(33,103,57,109)(34,84,58,126)(35,101,59,143)(36,82,60,124) );
G=PermutationGroup([(2,44),(4,46),(6,48),(8,50),(10,52),(12,54),(14,56),(16,58),(18,60),(20,62),(22,64),(24,66),(26,68),(28,70),(30,72),(32,38),(34,40),(36,42),(73,91),(74,116),(75,93),(76,118),(77,95),(78,120),(79,97),(80,122),(81,99),(82,124),(83,101),(84,126),(85,103),(86,128),(87,105),(88,130),(89,107),(90,132),(92,134),(94,136),(96,138),(98,140),(100,142),(102,144),(104,110),(106,112),(108,114),(109,127),(111,129),(113,131),(115,133),(117,135),(119,137),(121,139),(123,141),(125,143)], [(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)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42),(73,133),(74,134),(75,135),(76,136),(77,137),(78,138),(79,139),(80,140),(81,141),(82,142),(83,143),(84,144),(85,109),(86,110),(87,111),(88,112),(89,113),(90,114),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120),(97,121),(98,122),(99,123),(100,124),(101,125),(102,126),(103,127),(104,128),(105,129),(106,130),(107,131),(108,132)], [(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)], [(1,99,61,141),(2,80,62,122),(3,97,63,139),(4,78,64,120),(5,95,65,137),(6,76,66,118),(7,93,67,135),(8,74,68,116),(9,91,69,133),(10,108,70,114),(11,89,71,131),(12,106,72,112),(13,87,37,129),(14,104,38,110),(15,85,39,127),(16,102,40,144),(17,83,41,125),(18,100,42,142),(19,81,43,123),(20,98,44,140),(21,79,45,121),(22,96,46,138),(23,77,47,119),(24,94,48,136),(25,75,49,117),(26,92,50,134),(27,73,51,115),(28,90,52,132),(29,107,53,113),(30,88,54,130),(31,105,55,111),(32,86,56,128),(33,103,57,109),(34,84,58,126),(35,101,59,143),(36,82,60,124)])
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