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## G = D4×Dic9order 288 = 25·32

### Direct product of D4 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4×Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C22×Dic9 — D4×Dic9
 Lower central C9 — C18 — D4×Dic9
 Upper central C1 — C22 — C2×D4

Generators and relations for D4×Dic9
G = < a,b,c,d | a4=b2=c18=1, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 456 in 141 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C18, C18, C2×Dic3, C2×C12, C3×D4, C22×C6, C4×D4, Dic9, Dic9, C36, C2×C18, C2×C18, C2×C18, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C6×D4, C2×Dic9, C2×Dic9, C2×Dic9, C2×C36, D4×C9, C22×C18, D4×Dic3, C4×Dic9, C4⋊Dic9, C18.D4, C22×Dic9, D4×C18, D4×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, D9, C2×Dic3, C22×S3, C4×D4, Dic9, D18, S3×D4, D42S3, C22×Dic3, C2×Dic9, C22×D9, D4×Dic3, D4×D9, D42D9, C22×Dic9, D4×Dic9

Smallest permutation representation of D4×Dic9
On 144 points
Generators in S144
(1 38 141 123)(2 39 142 124)(3 40 143 125)(4 41 144 126)(5 42 127 109)(6 43 128 110)(7 44 129 111)(8 45 130 112)(9 46 131 113)(10 47 132 114)(11 48 133 115)(12 49 134 116)(13 50 135 117)(14 51 136 118)(15 52 137 119)(16 53 138 120)(17 54 139 121)(18 37 140 122)(19 57 105 84)(20 58 106 85)(21 59 107 86)(22 60 108 87)(23 61 91 88)(24 62 92 89)(25 63 93 90)(26 64 94 73)(27 65 95 74)(28 66 96 75)(29 67 97 76)(30 68 98 77)(31 69 99 78)(32 70 100 79)(33 71 101 80)(34 72 102 81)(35 55 103 82)(36 56 104 83)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 109)(15 110)(16 111)(17 112)(18 113)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 127)(52 128)(53 129)(54 130)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)
(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 72 10 63)(2 71 11 62)(3 70 12 61)(4 69 13 60)(5 68 14 59)(6 67 15 58)(7 66 16 57)(8 65 17 56)(9 64 18 55)(19 111 28 120)(20 110 29 119)(21 109 30 118)(22 126 31 117)(23 125 32 116)(24 124 33 115)(25 123 34 114)(26 122 35 113)(27 121 36 112)(37 103 46 94)(38 102 47 93)(39 101 48 92)(40 100 49 91)(41 99 50 108)(42 98 51 107)(43 97 52 106)(44 96 53 105)(45 95 54 104)(73 140 82 131)(74 139 83 130)(75 138 84 129)(76 137 85 128)(77 136 86 127)(78 135 87 144)(79 134 88 143)(80 133 89 142)(81 132 90 141)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 6 6 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 2 2 2 2 9 9 9 9 18 ··· 18 2 2 2 4 4 4 4 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + - + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 Dic3 D6 C4○D4 D9 D18 Dic9 D18 S3×D4 D4⋊2S3 D4×D9 D4⋊2D9 kernel D4×Dic9 C4×Dic9 C4⋊Dic9 C18.D4 C22×Dic9 D4×C18 D4×C9 C6×D4 Dic9 C2×C12 C3×D4 C22×C6 C18 C2×D4 C2×C4 D4 C23 C6 C6 C2 C2 # reps 1 1 1 2 2 1 8 1 2 1 4 2 2 3 3 12 6 1 1 3 3

Matrix representation of D4×Dic9 in GL5(𝔽37)

 36 0 0 0 0 0 1 35 0 0 0 1 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 36 0 0 0 0 0 1 35 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 20 0 0 0 17 31
,
 6 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 36 1

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,1,0,0,0,35,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,1,0,0,0,0,35,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,17,0,0,0,20,31],[6,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,36,0,0,0,0,1] >;

D4×Dic9 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_9
% in TeX

G:=Group("D4xDic9");
// GroupNames label

G:=SmallGroup(288,144);
// by ID

G=gap.SmallGroup(288,144);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,219,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^18=1,d^2=c^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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