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G = D4⋊Dic9order 288 = 25·32

1st semidirect product of D4 and Dic9 acting via Dic9/C18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36.8D4, D41Dic9, C18.12D8, C18.6SD16, (D4×C9)⋊1C4, C36.7(C2×C4), (C6×D4).2S3, (C2×D4).2D9, C93(D4⋊C4), C4⋊Dic910C2, C2.3(D4⋊D9), (D4×C18).2C2, (C2×C18).34D4, (C2×C4).41D18, (C2×C12).46D6, C4.1(C2×Dic9), C6.19(D4⋊S3), C4.13(C9⋊D4), C12.6(C3⋊D4), C2.3(D4.D9), C6.9(D4.S3), C3.(D4⋊Dic3), (C3×D4).1Dic3, C12.1(C2×Dic3), (C2×C36).24C22, C18.14(C22⋊C4), C22.17(C9⋊D4), C2.4(C18.D4), C6.15(C6.D4), (C2×C9⋊C8)⋊2C2, (C2×C6).72(C3⋊D4), SmallGroup(288,40)

Series: Derived Chief Lower central Upper central

C1C36 — D4⋊Dic9
C1C3C9C18C2×C18C2×C36C4⋊Dic9 — D4⋊Dic9
C9C18C36 — D4⋊Dic9
C1C22C2×C4C2×D4

Generators and relations for D4⋊Dic9
 G = < a,b,c,d | a4=b2=c18=1, d2=c9, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 248 in 75 conjugacy classes, 36 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C9, Dic3, C12 [×2], C2×C6, C2×C6 [×4], C4⋊C4, C2×C8, C2×D4, C18 [×3], C18 [×2], C3⋊C8, C2×Dic3, C2×C12, C3×D4 [×2], C3×D4, C22×C6, D4⋊C4, Dic9, C36 [×2], C2×C18, C2×C18 [×4], C2×C3⋊C8, C4⋊Dic3, C6×D4, C9⋊C8, C2×Dic9, C2×C36, D4×C9 [×2], D4×C9, C22×C18, D4⋊Dic3, C2×C9⋊C8, C4⋊Dic9, D4×C18, D4⋊Dic9
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D8, SD16, D9, C2×Dic3, C3⋊D4 [×2], D4⋊C4, Dic9 [×2], D18, D4⋊S3, D4.S3, C6.D4, C2×Dic9, C9⋊D4 [×2], D4⋊Dic3, D4.D9, D4⋊D9, C18.D4, D4⋊Dic9

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C6D6E6F6G8A8B8C8D9A9B9C12A12B18A···18I18J···18U36A···36F
order1222223444466666668888999121218···1818···1836···36
size1111442223636222444418181818222442···24···44···4

54 irreducible representations

dim11111222222222222224444
type++++++++-+++-+--+
imageC1C2C2C2C4S3D4D4D6Dic3D8SD16D9C3⋊D4C3⋊D4D18Dic9C9⋊D4C9⋊D4D4⋊S3D4.S3D4.D9D4⋊D9
kernelD4⋊Dic9C2×C9⋊C8C4⋊Dic9D4×C18D4×C9C6×D4C36C2×C18C2×C12C3×D4C18C18C2×D4C12C2×C6C2×C4D4C4C22C6C6C2C2
# reps11114111122232236661133

Matrix representation of D4⋊Dic9 in GL4(𝔽73) generated by

72000
07200
00129
001072
,
1000
167200
00129
00072
,
36000
127100
0010
0001
,
21800
207100
00026
00590
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,10,0,0,29,72],[1,16,0,0,0,72,0,0,0,0,1,0,0,0,29,72],[36,12,0,0,0,71,0,0,0,0,1,0,0,0,0,1],[2,20,0,0,18,71,0,0,0,0,0,59,0,0,26,0] >;

D4⋊Dic9 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,40);
// by ID

G=gap.SmallGroup(288,40);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,675,346,80,6725,292,9414]);
// Polycyclic

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

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