metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.16D18, Dic9⋊C4⋊7C2, (C2×Dic9)⋊3C4, (C4×Dic9)⋊9C2, (C2×C4).24D18, C22⋊C4.3D9, C22.6(C4×D9), C9⋊2(C42⋊C2), (C2×C12).174D6, C18.5(C22×C4), Dic9.6(C2×C4), (C22×C6).37D6, C18.20(C4○D4), C2.1(D4⋊2D9), (C2×C36).53C22, (C2×C18).18C23, C6.72(D4⋊2S3), C18.D4.1C2, (C22×C18).7C22, C3.(C23.16D6), (C22×Dic9).2C2, C22.12(C22×D9), (C2×Dic9).23C22, C2.7(C2×C4×D9), C6.44(S3×C2×C4), (C2×C6).5(C4×S3), (C2×C18).4(C2×C4), (C9×C22⋊C4).3C2, (C3×C22⋊C4).11S3, (C2×C6).175(C22×S3), SmallGroup(288,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.16D18
G = < a,b,c,d,e | a2=b2=c2=1, d18=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d17 >
Subgroups: 368 in 114 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C18, C18, C18, C2×Dic3, C2×C12, C22×C6, C42⋊C2, Dic9, Dic9, C36, C2×C18, C2×C18, C2×C18, C4×Dic3, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C2×Dic9, C2×Dic9, C2×C36, C22×C18, C23.16D6, C4×Dic9, Dic9⋊C4, C18.D4, C9×C22⋊C4, C22×Dic9, C23.16D18
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, D9, C4×S3, C22×S3, C42⋊C2, D18, S3×C2×C4, D4⋊2S3, C4×D9, C22×D9, C23.16D6, C2×C4×D9, D4⋊2D9, C23.16D18
►On 144 points
Generators in S144
(2 98)(4 100)(6 102)(8 104)(10 106)(12 108)(14 74)(16 76)(18 78)(20 80)(22 82)(24 84)(26 86)(28 88)(30 90)(32 92)(34 94)(36 96)(38 127)(40 129)(42 131)(44 133)(46 135)(48 137)(50 139)(52 141)(54 143)(56 109)(58 111)(60 113)(62 115)(64 117)(66 119)(68 121)(70 123)(72 125)
(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 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 126)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 141)(53 142)(54 143)(55 144)(56 109)(57 110)(58 111)(59 112)(60 113)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 121)(69 122)(70 123)(71 124)(72 125)
(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)
(1 117 79 46)(2 134 80 63)(3 115 81 44)(4 132 82 61)(5 113 83 42)(6 130 84 59)(7 111 85 40)(8 128 86 57)(9 109 87 38)(10 126 88 55)(11 143 89 72)(12 124 90 53)(13 141 91 70)(14 122 92 51)(15 139 93 68)(16 120 94 49)(17 137 95 66)(18 118 96 47)(19 135 97 64)(20 116 98 45)(21 133 99 62)(22 114 100 43)(23 131 101 60)(24 112 102 41)(25 129 103 58)(26 110 104 39)(27 127 105 56)(28 144 106 37)(29 125 107 54)(30 142 108 71)(31 123 73 52)(32 140 74 69)(33 121 75 50)(34 138 76 67)(35 119 77 48)(36 136 78 65)
G:=sub<Sym(144)| (2,98)(4,100)(6,102)(8,104)(10,106)(12,108)(14,74)(16,76)(18,78)(20,80)(22,82)(24,84)(26,86)(28,88)(30,90)(32,92)(34,94)(36,96)(38,127)(40,129)(42,131)(44,133)(46,135)(48,137)(50,139)(52,141)(54,143)(56,109)(58,111)(60,113)(62,115)(64,117)(66,119)(68,121)(70,123)(72,125), (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), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125), (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), (1,117,79,46)(2,134,80,63)(3,115,81,44)(4,132,82,61)(5,113,83,42)(6,130,84,59)(7,111,85,40)(8,128,86,57)(9,109,87,38)(10,126,88,55)(11,143,89,72)(12,124,90,53)(13,141,91,70)(14,122,92,51)(15,139,93,68)(16,120,94,49)(17,137,95,66)(18,118,96,47)(19,135,97,64)(20,116,98,45)(21,133,99,62)(22,114,100,43)(23,131,101,60)(24,112,102,41)(25,129,103,58)(26,110,104,39)(27,127,105,56)(28,144,106,37)(29,125,107,54)(30,142,108,71)(31,123,73,52)(32,140,74,69)(33,121,75,50)(34,138,76,67)(35,119,77,48)(36,136,78,65)>;
G:=Group( (2,98)(4,100)(6,102)(8,104)(10,106)(12,108)(14,74)(16,76)(18,78)(20,80)(22,82)(24,84)(26,86)(28,88)(30,90)(32,92)(34,94)(36,96)(38,127)(40,129)(42,131)(44,133)(46,135)(48,137)(50,139)(52,141)(54,143)(56,109)(58,111)(60,113)(62,115)(64,117)(66,119)(68,121)(70,123)(72,125), (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), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125), (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), (1,117,79,46)(2,134,80,63)(3,115,81,44)(4,132,82,61)(5,113,83,42)(6,130,84,59)(7,111,85,40)(8,128,86,57)(9,109,87,38)(10,126,88,55)(11,143,89,72)(12,124,90,53)(13,141,91,70)(14,122,92,51)(15,139,93,68)(16,120,94,49)(17,137,95,66)(18,118,96,47)(19,135,97,64)(20,116,98,45)(21,133,99,62)(22,114,100,43)(23,131,101,60)(24,112,102,41)(25,129,103,58)(26,110,104,39)(27,127,105,56)(28,144,106,37)(29,125,107,54)(30,142,108,71)(31,123,73,52)(32,140,74,69)(33,121,75,50)(34,138,76,67)(35,119,77,48)(36,136,78,65) );
G=PermutationGroup([[(2,98),(4,100),(6,102),(8,104),(10,106),(12,108),(14,74),(16,76),(18,78),(20,80),(22,82),(24,84),(26,86),(28,88),(30,90),(32,92),(34,94),(36,96),(38,127),(40,129),(42,131),(44,133),(46,135),(48,137),(50,139),(52,141),(54,143),(56,109),(58,111),(60,113),(62,115),(64,117),(66,119),(68,121),(70,123),(72,125)], [(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)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,106),(11,107),(12,108),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,126),(38,127),(39,128),(40,129),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,140),(52,141),(53,142),(54,143),(55,144),(56,109),(57,110),(58,111),(59,112),(60,113),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,121),(69,122),(70,123),(71,124),(72,125)], [(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)], [(1,117,79,46),(2,134,80,63),(3,115,81,44),(4,132,82,61),(5,113,83,42),(6,130,84,59),(7,111,85,40),(8,128,86,57),(9,109,87,38),(10,126,88,55),(11,143,89,72),(12,124,90,53),(13,141,91,70),(14,122,92,51),(15,139,93,68),(16,120,94,49),(17,137,95,66),(18,118,96,47),(19,135,97,64),(20,116,98,45),(21,133,99,62),(22,114,100,43),(23,131,101,60),(24,112,102,41),(25,129,103,58),(26,110,104,39),(27,127,105,56),(28,144,106,37),(29,125,107,54),(30,142,108,71),(31,123,73,52),(32,140,74,69),(33,121,75,50),(34,138,76,67),(35,119,77,48),(36,136,78,65)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 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 | 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 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 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 | C4 | S3 | D6 | D6 | C4○D4 | D9 | C4×S3 | D18 | D18 | C4×D9 | D4⋊2S3 | D4⋊2D9 |
| kernel | C23.16D18 | C4×Dic9 | Dic9⋊C4 | C18.D4 | C9×C22⋊C4 | C22×Dic9 | C2×Dic9 | C3×C22⋊C4 | C2×C12 | C22×C6 | C18 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 4 | 3 | 4 | 6 | 3 | 12 | 2 | 6 |
Matrix representation of C23.16D18 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 11 | 20 | 0 | 0 |
| 0 | 17 | 31 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 35 | 36 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 12 | 33 | 0 | 0 |
| 0 | 8 | 25 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 12 | 6 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,36],[36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[6,0,0,0,0,0,11,17,0,0,0,20,31,0,0,0,0,0,1,35,0,0,0,0,36],[6,0,0,0,0,0,12,8,0,0,0,33,25,0,0,0,0,0,31,12,0,0,0,0,6] >;
C23.16D18 in GAP, Magma, Sage, TeX
C_2^3._{16}D_{18} % in TeX
G:=Group("C2^3.16D18"); // GroupNames label
G:=SmallGroup(288,87);
// by ID
G=gap.SmallGroup(288,87);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^18=b,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-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