Copied to
clipboard

## G = D4⋊Dic9order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — D4⋊Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×C36 — C4⋊Dic9 — D4⋊Dic9
 Lower central C9 — C18 — C36 — D4⋊Dic9
 Upper central C1 — C22 — C2×C4 — C2×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 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 4 4 2 2 2 36 36 2 2 2 4 4 4 4 18 18 18 18 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + - + - - + image C1 C2 C2 C2 C4 S3 D4 D4 D6 Dic3 D8 SD16 D9 C3⋊D4 C3⋊D4 D18 Dic9 C9⋊D4 C9⋊D4 D4⋊S3 D4.S3 D4.D9 D4⋊D9 kernel D4⋊Dic9 C2×C9⋊C8 C4⋊Dic9 D4×C18 D4×C9 C6×D4 C36 C2×C18 C2×C12 C3×D4 C18 C18 C2×D4 C12 C2×C6 C2×C4 D4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 3 2 2 3 6 6 6 1 1 3 3

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

 72 0 0 0 0 72 0 0 0 0 1 29 0 0 10 72
,
 1 0 0 0 16 72 0 0 0 0 1 29 0 0 0 72
,
 36 0 0 0 12 71 0 0 0 0 1 0 0 0 0 1
,
 2 18 0 0 20 71 0 0 0 0 0 26 0 0 59 0
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

׿
×
𝔽