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## G = SD16×C3×C6order 288 = 25·32

### Direct product of C3×C6 and SD16

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C3×C6
 Chief series C1 — C2 — C4 — C12 — C3×C12 — Q8×C32 — C32×SD16 — SD16×C3×C6
 Lower central C1 — C2 — C4 — SD16×C3×C6
 Upper central C1 — C62 — C6×C12 — SD16×C3×C6

Generators and relations for SD16×C3×C6
G = < a,b,c,d | a3=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 324 in 204 conjugacy classes, 132 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C6 [×12], C6 [×8], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×2], Q8, C23, C32, C12 [×8], C12 [×8], C2×C6 [×4], C2×C6 [×16], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×C6, C3×C6 [×2], C3×C6 [×2], C24 [×8], C2×C12 [×4], C2×C12 [×4], C3×D4 [×8], C3×D4 [×4], C3×Q8 [×8], C3×Q8 [×4], C22×C6 [×4], C2×SD16, C3×C12 [×2], C3×C12 [×2], C62, C62 [×4], C2×C24 [×4], C3×SD16 [×16], C6×D4 [×4], C6×Q8 [×4], C3×C24 [×2], C6×C12, C6×C12, D4×C32 [×2], D4×C32, Q8×C32 [×2], Q8×C32, C2×C62, C6×SD16 [×4], C6×C24, C32×SD16 [×4], D4×C3×C6, Q8×C3×C6, SD16×C3×C6
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×2], C23, C32, C2×C6 [×28], SD16 [×2], C2×D4, C3×C6 [×7], C3×D4 [×8], C22×C6 [×4], C2×SD16, C62 [×7], C3×SD16 [×8], C6×D4 [×4], D4×C32 [×2], C2×C62, C6×SD16 [×4], C32×SD16 [×2], D4×C3×C6, SD16×C3×C6

Smallest permutation representation of SD16×C3×C6
On 144 points
Generators in S144
(1 26 18)(2 27 19)(3 28 20)(4 29 21)(5 30 22)(6 31 23)(7 32 24)(8 25 17)(9 33 130)(10 34 131)(11 35 132)(12 36 133)(13 37 134)(14 38 135)(15 39 136)(16 40 129)(41 90 49)(42 91 50)(43 92 51)(44 93 52)(45 94 53)(46 95 54)(47 96 55)(48 89 56)(57 77 65)(58 78 66)(59 79 67)(60 80 68)(61 73 69)(62 74 70)(63 75 71)(64 76 72)(81 121 113)(82 122 114)(83 123 115)(84 124 116)(85 125 117)(86 126 118)(87 127 119)(88 128 120)(97 137 105)(98 138 106)(99 139 107)(100 140 108)(101 141 109)(102 142 110)(103 143 111)(104 144 112)
(1 103 83 62 42 135)(2 104 84 63 43 136)(3 97 85 64 44 129)(4 98 86 57 45 130)(5 99 87 58 46 131)(6 100 88 59 47 132)(7 101 81 60 48 133)(8 102 82 61 41 134)(9 29 138 126 77 94)(10 30 139 127 78 95)(11 31 140 128 79 96)(12 32 141 121 80 89)(13 25 142 122 73 90)(14 26 143 123 74 91)(15 27 144 124 75 92)(16 28 137 125 76 93)(17 110 114 69 49 37)(18 111 115 70 50 38)(19 112 116 71 51 39)(20 105 117 72 52 40)(21 106 118 65 53 33)(22 107 119 66 54 34)(23 108 120 67 55 35)(24 109 113 68 56 36)
(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 58)(2 61)(3 64)(4 59)(5 62)(6 57)(7 60)(8 63)(9 128)(10 123)(11 126)(12 121)(13 124)(14 127)(15 122)(16 125)(17 71)(18 66)(19 69)(20 72)(21 67)(22 70)(23 65)(24 68)(25 75)(26 78)(27 73)(28 76)(29 79)(30 74)(31 77)(32 80)(33 120)(34 115)(35 118)(36 113)(37 116)(38 119)(39 114)(40 117)(41 104)(42 99)(43 102)(44 97)(45 100)(46 103)(47 98)(48 101)(49 112)(50 107)(51 110)(52 105)(53 108)(54 111)(55 106)(56 109)(81 133)(82 136)(83 131)(84 134)(85 129)(86 132)(87 135)(88 130)(89 141)(90 144)(91 139)(92 142)(93 137)(94 140)(95 143)(96 138)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 4A 4B 4C 4D 6A ··· 6X 6Y ··· 6AN 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12AF 24A ··· 24AF order 1 2 2 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 4 4 1 ··· 1 2 2 4 4 1 ··· 1 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 SD16 C3×D4 C3×D4 C3×SD16 kernel SD16×C3×C6 C6×C24 C32×SD16 D4×C3×C6 Q8×C3×C6 C6×SD16 C2×C24 C3×SD16 C6×D4 C6×Q8 C3×C12 C62 C3×C6 C12 C2×C6 C6 # reps 1 1 4 1 1 8 8 32 8 8 1 1 4 8 8 32

Matrix representation of SD16×C3×C6 in GL5(𝔽73)

 8 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 9 0 0 0 0 0 9 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 6 67 0 0 0 6 6
,
 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72

G:=sub<GL(5,GF(73))| [8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,67,6],[72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72] >;

SD16×C3×C6 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_3\times C_6
% in TeX

G:=Group("SD16xC3xC6");
// GroupNames label

G:=SmallGroup(288,830);
// by ID

G=gap.SmallGroup(288,830);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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