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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

Aliases: SD16×C18, C36.43D4, C7213C22, C36.45C23, C83(C2×C18), (C2×C8)⋊5C18, C4.7(D4×C9), (C2×C72)⋊13C2, C3.(C6×SD16), Q82(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

C1C4 — SD16×C18
C1C2C6C12C36Q8×C9C9×SD16 — SD16×C18
C1C2C4 — SD16×C18
C1C2×C18C2×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 2A2B2C2D2E3A3B4A4B4C4D6A···6F6G6H6I6J8A8B8C8D9A···9F12A12B12C12D12E12F12G12H18A···18R18S···18AD24A···24H36A···36L36M···36X72A···72X
order1222223344446···6666688889···9121212121212121218···1818···1824···2436···3636···3672···72
size1111441122441···1444422221···1222244441···14···42···22···24···42···2

126 irreducible representations

dim111111111111111222222222
type+++++++
imageC1C2C2C2C2C3C6C6C6C6C9C18C18C18C18D4D4SD16C3×D4C3×D4C3×SD16D4×C9D4×C9C9×SD16
kernelSD16×C18C2×C72C9×SD16D4×C18Q8×C18C6×SD16C2×C24C3×SD16C6×D4C6×Q8C2×SD16C2×C8SD16C2×D4C2×Q8C36C2×C18C18C12C2×C6C6C4C22C2
# reps11411228226624661142286624

Matrix representation of SD16×C18 in GL4(𝔽73) generated by

18000
01800
0020
0002
,
552000
21800
00676
006767
,
11800
07200
0010
00072
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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