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G = Dic9⋊D4order 288 = 25·32

2nd semidirect product of Dic9 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic93D4, C23.14D18, (C2×D4)⋊5D9, (C2×C18)⋊3D4, (D4×C18)⋊9C2, C95(C4⋊D4), C2.27(D4×D9), D18⋊C415C2, (C6×D4).21S3, (C2×C4).16D18, C18.38(C2×D4), C6.102(S3×D4), Dic9⋊C415C2, (C2×C12).217D6, C222(C9⋊D4), (C22×C6).52D6, C18.32(C4○D4), C3.(C23.14D6), (C2×C18).54C23, (C2×C36).61C22, (C22×Dic9)⋊6C2, C18.D412C2, C2.18(D42D9), C6.89(D42S3), C22.61(C22×D9), (C22×C18).21C22, (C2×Dic9).40C22, (C22×D9).11C22, (C2×C9⋊D4)⋊6C2, C6.99(C2×C3⋊D4), C2.15(C2×C9⋊D4), (C2×C6).6(C3⋊D4), (C2×C6).211(C22×S3), SmallGroup(288,149)

Series: Derived Chief Lower central Upper central

C1C2×C18 — Dic9⋊D4
C1C3C9C18C2×C18C22×D9C2×C9⋊D4 — Dic9⋊D4
C9C2×C18 — Dic9⋊D4
C1C22C2×D4

Generators and relations for Dic9⋊D4
 G = < a,b,c,d | a18=c4=d2=1, b2=a9, bab-1=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd=c-1 >

Subgroups: 612 in 141 conjugacy classes, 46 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×2], C22 [×8], S3, C6 [×3], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C9, Dic3 [×4], C12, D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], D9, C18 [×3], C18 [×3], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C4⋊D4, Dic9 [×2], Dic9 [×2], C36, D18 [×3], C2×C18, C2×C18 [×2], C2×C18 [×5], Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C2×Dic9 [×3], C2×Dic9 [×2], C9⋊D4 [×4], C2×C36, D4×C9 [×2], C22×D9, C22×C18 [×2], C23.14D6, Dic9⋊C4, D18⋊C4, C18.D4, C22×Dic9, C2×C9⋊D4 [×2], D4×C18, Dic9⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D9, C3⋊D4 [×2], C22×S3, C4⋊D4, D18 [×3], S3×D4, D42S3, C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C23.14D6, D4×D9, D42D9, C2×C9⋊D4, Dic9⋊D4

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F6A6B6C6D6E6F6G9A9B9C12A12B18A···18I18J···18U36A···36F
order1222222234444446666666999121218···1818···1836···36
size1111224362418181818362224444222442···24···44···4

54 irreducible representations

dim1111111222222222224444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2S3D4D4D6D6C4○D4D9C3⋊D4D18D18C9⋊D4S3×D4D42S3D4×D9D42D9
kernelDic9⋊D4Dic9⋊C4D18⋊C4C18.D4C22×Dic9C2×C9⋊D4D4×C18C6×D4Dic9C2×C18C2×C12C22×C6C18C2×D4C2×C6C2×C4C23C22C6C6C2C2
# reps11111211221223436121133

Matrix representation of Dic9⋊D4 in GL4(𝔽37) generated by

173100
61100
0010
0001
,
21100
133500
00360
00036
,
72300
143000
003635
0011
,
36000
03600
003635
0001
G:=sub<GL(4,GF(37))| [17,6,0,0,31,11,0,0,0,0,1,0,0,0,0,1],[2,13,0,0,11,35,0,0,0,0,36,0,0,0,0,36],[7,14,0,0,23,30,0,0,0,0,36,1,0,0,35,1],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,35,1] >;

Dic9⋊D4 in GAP, Magma, Sage, TeX

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

G:=Group("Dic9:D4");
// GroupNames label

G:=SmallGroup(288,149);
// by ID

G=gap.SmallGroup(288,149);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,6725,292,9414]);
// Polycyclic

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

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