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## G = SD16×C18order 288 = 25·32

### Direct product of C18 and SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C18
 Chief series C1 — C2 — C6 — C12 — C36 — Q8×C9 — C9×SD16 — SD16×C18
 Lower central C1 — C2 — C4 — SD16×C18
 Upper central C1 — C2×C18 — C2×C36 — SD16×C18

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 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×2], Q8, C23, C9, C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C2×C8, SD16 [×4], C2×D4, C2×Q8, C18, C18 [×2], C18 [×2], C24 [×2], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×2], C3×Q8, C22×C6, C2×SD16, C36 [×2], C36 [×2], C2×C18, C2×C18 [×4], C2×C24, C3×SD16 [×4], C6×D4, C6×Q8, C72 [×2], C2×C36, C2×C36, D4×C9 [×2], D4×C9, Q8×C9 [×2], Q8×C9, C22×C18, C6×SD16, C2×C72, C9×SD16 [×4], D4×C18, Q8×C18, SD16×C18
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], SD16 [×2], C2×D4, C18 [×7], C3×D4 [×2], C22×C6, C2×SD16, C2×C18 [×7], C3×SD16 [×2], C6×D4, D4×C9 [×2], C22×C18, C6×SD16, C9×SD16 [×2], D4×C18, SD16×C18

Smallest permutation representation of 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 59 131 104 49 22 89 122)(2 60 132 105 50 23 90 123)(3 61 133 106 51 24 73 124)(4 62 134 107 52 25 74 125)(5 63 135 108 53 26 75 126)(6 64 136 91 54 27 76 109)(7 65 137 92 37 28 77 110)(8 66 138 93 38 29 78 111)(9 67 139 94 39 30 79 112)(10 68 140 95 40 31 80 113)(11 69 141 96 41 32 81 114)(12 70 142 97 42 33 82 115)(13 71 143 98 43 34 83 116)(14 72 144 99 44 35 84 117)(15 55 127 100 45 36 85 118)(16 56 128 101 46 19 86 119)(17 57 129 102 47 20 87 120)(18 58 130 103 48 21 88 121)
(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(61 106)(62 107)(63 108)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)(82 142)(83 143)(84 144)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)

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), (1,59,131,104,49,22,89,122)(2,60,132,105,50,23,90,123)(3,61,133,106,51,24,73,124)(4,62,134,107,52,25,74,125)(5,63,135,108,53,26,75,126)(6,64,136,91,54,27,76,109)(7,65,137,92,37,28,77,110)(8,66,138,93,38,29,78,111)(9,67,139,94,39,30,79,112)(10,68,140,95,40,31,80,113)(11,69,141,96,41,32,81,114)(12,70,142,97,42,33,82,115)(13,71,143,98,43,34,83,116)(14,72,144,99,44,35,84,117)(15,55,127,100,45,36,85,118)(16,56,128,101,46,19,86,119)(17,57,129,102,47,20,87,120)(18,58,130,103,48,21,88,121), (19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(83,143)(84,144)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)>;

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), (1,59,131,104,49,22,89,122)(2,60,132,105,50,23,90,123)(3,61,133,106,51,24,73,124)(4,62,134,107,52,25,74,125)(5,63,135,108,53,26,75,126)(6,64,136,91,54,27,76,109)(7,65,137,92,37,28,77,110)(8,66,138,93,38,29,78,111)(9,67,139,94,39,30,79,112)(10,68,140,95,40,31,80,113)(11,69,141,96,41,32,81,114)(12,70,142,97,42,33,82,115)(13,71,143,98,43,34,83,116)(14,72,144,99,44,35,84,117)(15,55,127,100,45,36,85,118)(16,56,128,101,46,19,86,119)(17,57,129,102,47,20,87,120)(18,58,130,103,48,21,88,121), (19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(83,143)(84,144)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132) );

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)], [(1,59,131,104,49,22,89,122),(2,60,132,105,50,23,90,123),(3,61,133,106,51,24,73,124),(4,62,134,107,52,25,74,125),(5,63,135,108,53,26,75,126),(6,64,136,91,54,27,76,109),(7,65,137,92,37,28,77,110),(8,66,138,93,38,29,78,111),(9,67,139,94,39,30,79,112),(10,68,140,95,40,31,80,113),(11,69,141,96,41,32,81,114),(12,70,142,97,42,33,82,115),(13,71,143,98,43,34,83,116),(14,72,144,99,44,35,84,117),(15,55,127,100,45,36,85,118),(16,56,128,101,46,19,86,119),(17,57,129,102,47,20,87,120),(18,58,130,103,48,21,88,121)], [(19,119),(20,120),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(55,100),(56,101),(57,102),(58,103),(59,104),(60,105),(61,106),(62,107),(63,108),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,133),(74,134),(75,135),(76,136),(77,137),(78,138),(79,139),(80,140),(81,141),(82,142),(83,143),(84,144),(85,127),(86,128),(87,129),(88,130),(89,131),(90,132)])

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

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