Aliases: C18.5S4, C9⋊CSU2(𝔽3), SL2(𝔽3).D9, Q8⋊C9.1S3, C6.1(C3⋊S4), C2.2(C9⋊S4), (Q8×C9).3S3, Q8.1(C9⋊S3), C3.1(C6.5S4), (C9×SL2(𝔽3)).1C2, (C3×SL2(𝔽3)).4S3, (C3×Q8).1(C3⋊S3), SmallGroup(432,252)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C9×SL2(𝔽3) — C18.5S4 |
| C9×SL2(𝔽3) — C18.5S4 |
Generators and relations for C18.5S4
G = < a,b,c,d,e | a18=d3=1, b2=c2=e2=a9, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a9b, dbd-1=a9bc, ebe-1=bc, dcd-1=b, ece-1=a9c, ede-1=d-1 >
Subgroups: 461 in 59 conjugacy classes, 17 normal (12 characteristic)
C1, C2, C3, C3, C4, C6, C6, C8, Q8, Q8, C9, C9, C32, Dic3, C12, Q16, C18, C18, C3×C6, C3⋊C8, SL2(𝔽3), Dic6, C3×Q8, C3×C9, Dic9, C36, C3⋊Dic3, C3⋊Q16, CSU2(𝔽3), C3×C18, C9⋊C8, Q8⋊C9, Dic18, Q8×C9, C3×SL2(𝔽3), C9⋊Dic3, C9⋊Q16, Q8.D9, C6.5S4, C9×SL2(𝔽3), C18.5S4
Quotients: C1, C2, S3, D9, C3⋊S3, S4, CSU2(𝔽3), C9⋊S3, C3⋊S4, C6.5S4, C9⋊S4, C18.5S4
►On 144 points
Generators in S144
(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)
(1 108 10 99)(2 91 11 100)(3 92 12 101)(4 93 13 102)(5 94 14 103)(6 95 15 104)(7 96 16 105)(8 97 17 106)(9 98 18 107)(19 53 28 44)(20 54 29 45)(21 37 30 46)(22 38 31 47)(23 39 32 48)(24 40 33 49)(25 41 34 50)(26 42 35 51)(27 43 36 52)(55 139 64 130)(56 140 65 131)(57 141 66 132)(58 142 67 133)(59 143 68 134)(60 144 69 135)(61 127 70 136)(62 128 71 137)(63 129 72 138)(73 117 82 126)(74 118 83 109)(75 119 84 110)(76 120 85 111)(77 121 86 112)(78 122 87 113)(79 123 88 114)(80 124 89 115)(81 125 90 116)
(1 110 10 119)(2 111 11 120)(3 112 12 121)(4 113 13 122)(5 114 14 123)(6 115 15 124)(7 116 16 125)(8 117 17 126)(9 118 18 109)(19 61 28 70)(20 62 29 71)(21 63 30 72)(22 64 31 55)(23 65 32 56)(24 66 33 57)(25 67 34 58)(26 68 35 59)(27 69 36 60)(37 138 46 129)(38 139 47 130)(39 140 48 131)(40 141 49 132)(41 142 50 133)(42 143 51 134)(43 144 52 135)(44 127 53 136)(45 128 54 137)(73 106 82 97)(74 107 83 98)(75 108 84 99)(76 91 85 100)(77 92 86 101)(78 93 87 102)(79 94 88 103)(80 95 89 104)(81 96 90 105)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 67 38)(20 68 39)(21 69 40)(22 70 41)(23 71 42)(24 72 43)(25 55 44)(26 56 45)(27 57 46)(28 58 47)(29 59 48)(30 60 49)(31 61 50)(32 62 51)(33 63 52)(34 64 53)(35 65 54)(36 66 37)(73 94 120)(74 95 121)(75 96 122)(76 97 123)(77 98 124)(78 99 125)(79 100 126)(80 101 109)(81 102 110)(82 103 111)(83 104 112)(84 105 113)(85 106 114)(86 107 115)(87 108 116)(88 91 117)(89 92 118)(90 93 119)(127 133 139)(128 134 140)(129 135 141)(130 136 142)(131 137 143)(132 138 144)
(1 132 10 141)(2 131 11 140)(3 130 12 139)(4 129 13 138)(5 128 14 137)(6 127 15 136)(7 144 16 135)(8 143 17 134)(9 142 18 133)(19 104 28 95)(20 103 29 94)(21 102 30 93)(22 101 31 92)(23 100 32 91)(24 99 33 108)(25 98 34 107)(26 97 35 106)(27 96 36 105)(37 113 46 122)(38 112 47 121)(39 111 48 120)(40 110 49 119)(41 109 50 118)(42 126 51 117)(43 125 52 116)(44 124 53 115)(45 123 54 114)(55 77 64 86)(56 76 65 85)(57 75 66 84)(58 74 67 83)(59 73 68 82)(60 90 69 81)(61 89 70 80)(62 88 71 79)(63 87 72 78)
G:=sub<Sym(144)| (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), (1,108,10,99)(2,91,11,100)(3,92,12,101)(4,93,13,102)(5,94,14,103)(6,95,15,104)(7,96,16,105)(8,97,17,106)(9,98,18,107)(19,53,28,44)(20,54,29,45)(21,37,30,46)(22,38,31,47)(23,39,32,48)(24,40,33,49)(25,41,34,50)(26,42,35,51)(27,43,36,52)(55,139,64,130)(56,140,65,131)(57,141,66,132)(58,142,67,133)(59,143,68,134)(60,144,69,135)(61,127,70,136)(62,128,71,137)(63,129,72,138)(73,117,82,126)(74,118,83,109)(75,119,84,110)(76,120,85,111)(77,121,86,112)(78,122,87,113)(79,123,88,114)(80,124,89,115)(81,125,90,116), (1,110,10,119)(2,111,11,120)(3,112,12,121)(4,113,13,122)(5,114,14,123)(6,115,15,124)(7,116,16,125)(8,117,17,126)(9,118,18,109)(19,61,28,70)(20,62,29,71)(21,63,30,72)(22,64,31,55)(23,65,32,56)(24,66,33,57)(25,67,34,58)(26,68,35,59)(27,69,36,60)(37,138,46,129)(38,139,47,130)(39,140,48,131)(40,141,49,132)(41,142,50,133)(42,143,51,134)(43,144,52,135)(44,127,53,136)(45,128,54,137)(73,106,82,97)(74,107,83,98)(75,108,84,99)(76,91,85,100)(77,92,86,101)(78,93,87,102)(79,94,88,103)(80,95,89,104)(81,96,90,105), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,67,38)(20,68,39)(21,69,40)(22,70,41)(23,71,42)(24,72,43)(25,55,44)(26,56,45)(27,57,46)(28,58,47)(29,59,48)(30,60,49)(31,61,50)(32,62,51)(33,63,52)(34,64,53)(35,65,54)(36,66,37)(73,94,120)(74,95,121)(75,96,122)(76,97,123)(77,98,124)(78,99,125)(79,100,126)(80,101,109)(81,102,110)(82,103,111)(83,104,112)(84,105,113)(85,106,114)(86,107,115)(87,108,116)(88,91,117)(89,92,118)(90,93,119)(127,133,139)(128,134,140)(129,135,141)(130,136,142)(131,137,143)(132,138,144), (1,132,10,141)(2,131,11,140)(3,130,12,139)(4,129,13,138)(5,128,14,137)(6,127,15,136)(7,144,16,135)(8,143,17,134)(9,142,18,133)(19,104,28,95)(20,103,29,94)(21,102,30,93)(22,101,31,92)(23,100,32,91)(24,99,33,108)(25,98,34,107)(26,97,35,106)(27,96,36,105)(37,113,46,122)(38,112,47,121)(39,111,48,120)(40,110,49,119)(41,109,50,118)(42,126,51,117)(43,125,52,116)(44,124,53,115)(45,123,54,114)(55,77,64,86)(56,76,65,85)(57,75,66,84)(58,74,67,83)(59,73,68,82)(60,90,69,81)(61,89,70,80)(62,88,71,79)(63,87,72,78)>;
G:=Group( (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), (1,108,10,99)(2,91,11,100)(3,92,12,101)(4,93,13,102)(5,94,14,103)(6,95,15,104)(7,96,16,105)(8,97,17,106)(9,98,18,107)(19,53,28,44)(20,54,29,45)(21,37,30,46)(22,38,31,47)(23,39,32,48)(24,40,33,49)(25,41,34,50)(26,42,35,51)(27,43,36,52)(55,139,64,130)(56,140,65,131)(57,141,66,132)(58,142,67,133)(59,143,68,134)(60,144,69,135)(61,127,70,136)(62,128,71,137)(63,129,72,138)(73,117,82,126)(74,118,83,109)(75,119,84,110)(76,120,85,111)(77,121,86,112)(78,122,87,113)(79,123,88,114)(80,124,89,115)(81,125,90,116), (1,110,10,119)(2,111,11,120)(3,112,12,121)(4,113,13,122)(5,114,14,123)(6,115,15,124)(7,116,16,125)(8,117,17,126)(9,118,18,109)(19,61,28,70)(20,62,29,71)(21,63,30,72)(22,64,31,55)(23,65,32,56)(24,66,33,57)(25,67,34,58)(26,68,35,59)(27,69,36,60)(37,138,46,129)(38,139,47,130)(39,140,48,131)(40,141,49,132)(41,142,50,133)(42,143,51,134)(43,144,52,135)(44,127,53,136)(45,128,54,137)(73,106,82,97)(74,107,83,98)(75,108,84,99)(76,91,85,100)(77,92,86,101)(78,93,87,102)(79,94,88,103)(80,95,89,104)(81,96,90,105), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,67,38)(20,68,39)(21,69,40)(22,70,41)(23,71,42)(24,72,43)(25,55,44)(26,56,45)(27,57,46)(28,58,47)(29,59,48)(30,60,49)(31,61,50)(32,62,51)(33,63,52)(34,64,53)(35,65,54)(36,66,37)(73,94,120)(74,95,121)(75,96,122)(76,97,123)(77,98,124)(78,99,125)(79,100,126)(80,101,109)(81,102,110)(82,103,111)(83,104,112)(84,105,113)(85,106,114)(86,107,115)(87,108,116)(88,91,117)(89,92,118)(90,93,119)(127,133,139)(128,134,140)(129,135,141)(130,136,142)(131,137,143)(132,138,144), (1,132,10,141)(2,131,11,140)(3,130,12,139)(4,129,13,138)(5,128,14,137)(6,127,15,136)(7,144,16,135)(8,143,17,134)(9,142,18,133)(19,104,28,95)(20,103,29,94)(21,102,30,93)(22,101,31,92)(23,100,32,91)(24,99,33,108)(25,98,34,107)(26,97,35,106)(27,96,36,105)(37,113,46,122)(38,112,47,121)(39,111,48,120)(40,110,49,119)(41,109,50,118)(42,126,51,117)(43,125,52,116)(44,124,53,115)(45,123,54,114)(55,77,64,86)(56,76,65,85)(57,75,66,84)(58,74,67,83)(59,73,68,82)(60,90,69,81)(61,89,70,80)(62,88,71,79)(63,87,72,78) );
G=PermutationGroup([[(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)], [(1,108,10,99),(2,91,11,100),(3,92,12,101),(4,93,13,102),(5,94,14,103),(6,95,15,104),(7,96,16,105),(8,97,17,106),(9,98,18,107),(19,53,28,44),(20,54,29,45),(21,37,30,46),(22,38,31,47),(23,39,32,48),(24,40,33,49),(25,41,34,50),(26,42,35,51),(27,43,36,52),(55,139,64,130),(56,140,65,131),(57,141,66,132),(58,142,67,133),(59,143,68,134),(60,144,69,135),(61,127,70,136),(62,128,71,137),(63,129,72,138),(73,117,82,126),(74,118,83,109),(75,119,84,110),(76,120,85,111),(77,121,86,112),(78,122,87,113),(79,123,88,114),(80,124,89,115),(81,125,90,116)], [(1,110,10,119),(2,111,11,120),(3,112,12,121),(4,113,13,122),(5,114,14,123),(6,115,15,124),(7,116,16,125),(8,117,17,126),(9,118,18,109),(19,61,28,70),(20,62,29,71),(21,63,30,72),(22,64,31,55),(23,65,32,56),(24,66,33,57),(25,67,34,58),(26,68,35,59),(27,69,36,60),(37,138,46,129),(38,139,47,130),(39,140,48,131),(40,141,49,132),(41,142,50,133),(42,143,51,134),(43,144,52,135),(44,127,53,136),(45,128,54,137),(73,106,82,97),(74,107,83,98),(75,108,84,99),(76,91,85,100),(77,92,86,101),(78,93,87,102),(79,94,88,103),(80,95,89,104),(81,96,90,105)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,67,38),(20,68,39),(21,69,40),(22,70,41),(23,71,42),(24,72,43),(25,55,44),(26,56,45),(27,57,46),(28,58,47),(29,59,48),(30,60,49),(31,61,50),(32,62,51),(33,63,52),(34,64,53),(35,65,54),(36,66,37),(73,94,120),(74,95,121),(75,96,122),(76,97,123),(77,98,124),(78,99,125),(79,100,126),(80,101,109),(81,102,110),(82,103,111),(83,104,112),(84,105,113),(85,106,114),(86,107,115),(87,108,116),(88,91,117),(89,92,118),(90,93,119),(127,133,139),(128,134,140),(129,135,141),(130,136,142),(131,137,143),(132,138,144)], [(1,132,10,141),(2,131,11,140),(3,130,12,139),(4,129,13,138),(5,128,14,137),(6,127,15,136),(7,144,16,135),(8,143,17,134),(9,142,18,133),(19,104,28,95),(20,103,29,94),(21,102,30,93),(22,101,31,92),(23,100,32,91),(24,99,33,108),(25,98,34,107),(26,97,35,106),(27,96,36,105),(37,113,46,122),(38,112,47,121),(39,111,48,120),(40,110,49,119),(41,109,50,118),(42,126,51,117),(43,125,52,116),(44,124,53,115),(45,123,54,114),(55,77,64,86),(56,76,65,85),(57,75,66,84),(58,74,67,83),(59,73,68,82),(60,90,69,81),(61,89,70,80),(62,88,71,79),(63,87,72,78)]])
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 9A | 9B | 9C | 9D | ··· | 9I | 12 | 18A | 18B | 18C | 18D | ··· | 18I | 36A | 36B | 36C |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 9 | ··· | 9 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | 36 | 36 |
| size | 1 | 1 | 2 | 8 | 8 | 8 | 6 | 108 | 2 | 8 | 8 | 8 | 54 | 54 | 2 | 2 | 2 | 8 | ··· | 8 | 12 | 2 | 2 | 2 | 8 | ··· | 8 | 12 | 12 | 12 |
36 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | - | + | - | - | - | + | + |
| image | C1 | C2 | S3 | S3 | S3 | D9 | CSU2(𝔽3) | S4 | CSU2(𝔽3) | C6.5S4 | C18.5S4 | C3⋊S4 | C9⋊S4 |
| kernel | C18.5S4 | C9×SL2(𝔽3) | Q8⋊C9 | Q8×C9 | C3×SL2(𝔽3) | SL2(𝔽3) | C9 | C18 | C9 | C3 | C1 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 9 | 2 | 2 | 1 | 3 | 9 | 1 | 3 |
Matrix representation of C18.5S4 ►in GL4(𝔽73) generated by
| 31 | 70 | 0 | 0 |
| 3 | 28 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 25 |
| 0 | 0 | 37 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 24 | 36 |
| 0 | 0 | 59 | 49 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 54 | 44 | 0 | 0 |
| 25 | 19 | 0 | 0 |
| 0 | 0 | 60 | 43 |
| 0 | 0 | 30 | 13 |
G:=sub<GL(4,GF(73))| [31,3,0,0,70,28,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,13,37,0,0,25,60],[1,0,0,0,0,1,0,0,0,0,24,59,0,0,36,49],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[54,25,0,0,44,19,0,0,0,0,60,30,0,0,43,13] >;
C18.5S4 in GAP, Magma, Sage, TeX
C_{18}._5S_4 % in TeX
G:=Group("C18.5S4"); // GroupNames label
G:=SmallGroup(432,252);
// by ID
G=gap.SmallGroup(432,252);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,57,632,142,1011,3784,5681,172,2273,3414,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^18=d^3=1,b^2=c^2=e^2=a^9,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^9*b,d*b*d^-1=a^9*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^9*c,e*d*e^-1=d^-1>;
// generators/relations