direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C18, C36.43D4, C72⋊13C22, C36.45C23, C8⋊3(C2×C18), (C2×C8)⋊5C18, C4.7(D4×C9), (C2×C72)⋊13C2, C3.(C6×SD16), Q8⋊2(C2×C18), (C2×Q8)⋊5C18, C6.75(C6×D4), (C6×SD16).C3, (C2×C24).26C6, C24.37(C2×C6), (Q8×C18)⋊10C2, (C6×D4).15C6, D4.1(C2×C18), (C2×D4).6C18, C2.12(D4×C18), C12.43(C3×D4), C18.75(C2×D4), (C2×C18).53D4, (C6×Q8).19C6, (Q8×C9)⋊9C22, (D4×C18).13C2, C4.2(C22×C18), (C3×SD16).3C6, C6.12(C3×SD16), C22.15(D4×C9), C12.45(C22×C6), (D4×C9).11C22, (C2×C36).128C22, (C2×C6).62(C3×D4), (C2×C4).27(C2×C18), (C3×D4).12(C2×C6), (C3×Q8).24(C2×C6), (C2×C12).145(C2×C6), SmallGroup(288,183)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C18
G = < a,b,c | a18=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 162 in 102 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C9, C12, C12, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C18, C18, C18, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C2×SD16, C36, C36, C2×C18, C2×C18, C2×C24, C3×SD16, C6×D4, C6×Q8, C72, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, Q8×C9, C22×C18, C6×SD16, C2×C72, C9×SD16, D4×C18, Q8×C18, SD16×C18
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, SD16, C2×D4, C18, C3×D4, C22×C6, C2×SD16, C2×C18, C3×SD16, C6×D4, D4×C9, C22×C18, C6×SD16, C9×SD16, D4×C18, SD16×C18
►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 93 32 78 128 72 52 114)(2 94 33 79 129 55 53 115)(3 95 34 80 130 56 54 116)(4 96 35 81 131 57 37 117)(5 97 36 82 132 58 38 118)(6 98 19 83 133 59 39 119)(7 99 20 84 134 60 40 120)(8 100 21 85 135 61 41 121)(9 101 22 86 136 62 42 122)(10 102 23 87 137 63 43 123)(11 103 24 88 138 64 44 124)(12 104 25 89 139 65 45 125)(13 105 26 90 140 66 46 126)(14 106 27 73 141 67 47 109)(15 107 28 74 142 68 48 110)(16 108 29 75 143 69 49 111)(17 91 30 76 144 70 50 112)(18 92 31 77 127 71 51 113)
(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 37)(36 38)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 106)(74 107)(75 108)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)
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), (1,93,32,78,128,72,52,114)(2,94,33,79,129,55,53,115)(3,95,34,80,130,56,54,116)(4,96,35,81,131,57,37,117)(5,97,36,82,132,58,38,118)(6,98,19,83,133,59,39,119)(7,99,20,84,134,60,40,120)(8,100,21,85,135,61,41,121)(9,101,22,86,136,62,42,122)(10,102,23,87,137,63,43,123)(11,103,24,88,138,64,44,124)(12,104,25,89,139,65,45,125)(13,105,26,90,140,66,46,126)(14,106,27,73,141,67,47,109)(15,107,28,74,142,68,48,110)(16,108,29,75,143,69,49,111)(17,91,30,76,144,70,50,112)(18,92,31,77,127,71,51,113), (19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,37)(36,38)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114)(73,106)(74,107)(75,108)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)>;
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), (1,93,32,78,128,72,52,114)(2,94,33,79,129,55,53,115)(3,95,34,80,130,56,54,116)(4,96,35,81,131,57,37,117)(5,97,36,82,132,58,38,118)(6,98,19,83,133,59,39,119)(7,99,20,84,134,60,40,120)(8,100,21,85,135,61,41,121)(9,101,22,86,136,62,42,122)(10,102,23,87,137,63,43,123)(11,103,24,88,138,64,44,124)(12,104,25,89,139,65,45,125)(13,105,26,90,140,66,46,126)(14,106,27,73,141,67,47,109)(15,107,28,74,142,68,48,110)(16,108,29,75,143,69,49,111)(17,91,30,76,144,70,50,112)(18,92,31,77,127,71,51,113), (19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,37)(36,38)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114)(73,106)(74,107)(75,108)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105) );
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)], [(1,93,32,78,128,72,52,114),(2,94,33,79,129,55,53,115),(3,95,34,80,130,56,54,116),(4,96,35,81,131,57,37,117),(5,97,36,82,132,58,38,118),(6,98,19,83,133,59,39,119),(7,99,20,84,134,60,40,120),(8,100,21,85,135,61,41,121),(9,101,22,86,136,62,42,122),(10,102,23,87,137,63,43,123),(11,103,24,88,138,64,44,124),(12,104,25,89,139,65,45,125),(13,105,26,90,140,66,46,126),(14,106,27,73,141,67,47,109),(15,107,28,74,142,68,48,110),(16,108,29,75,143,69,49,111),(17,91,30,76,144,70,50,112),(18,92,31,77,127,71,51,113)], [(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,37),(36,38),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,109),(68,110),(69,111),(70,112),(71,113),(72,114),(73,106),(74,107),(75,108),(76,91),(77,92),(78,93),(79,94),(80,95),(81,96),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18R | 18S | ··· | 18AD | 24A | ··· | 24H | 36A | ··· | 36L | 36M | ··· | 36X | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | D4 | D4 | SD16 | C3×D4 | C3×D4 | C3×SD16 | D4×C9 | D4×C9 | C9×SD16 |
| kernel | SD16×C18 | C2×C72 | C9×SD16 | D4×C18 | Q8×C18 | C6×SD16 | C2×C24 | C3×SD16 | C6×D4 | C6×Q8 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C36 | C2×C18 | C18 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 24 | 6 | 6 | 1 | 1 | 4 | 2 | 2 | 8 | 6 | 6 | 24 |
Matrix representation of SD16×C18 ►in GL4(𝔽73) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 55 | 20 | 0 | 0 |
| 2 | 18 | 0 | 0 |
| 0 | 0 | 67 | 6 |
| 0 | 0 | 67 | 67 |
| 1 | 18 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [18,0,0,0,0,18,0,0,0,0,2,0,0,0,0,2],[55,2,0,0,20,18,0,0,0,0,67,67,0,0,6,67],[1,0,0,0,18,72,0,0,0,0,1,0,0,0,0,72] >;
SD16×C18 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{18} % in TeX
G:=Group("SD16xC18"); // GroupNames label
G:=SmallGroup(288,183);
// by ID
G=gap.SmallGroup(288,183);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,1008,365,192,5884,2951,242]);
// Polycyclic
G:=Group<a,b,c|a^18=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations