Copied to
clipboard

## G = Dic9⋊4D4order 288 = 25·32

### 1st semidirect product of Dic9 and D4 acting through Inn(Dic9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — Dic9⋊4D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×C9⋊D4 — Dic9⋊4D4
 Lower central C9 — C18 — Dic9⋊4D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic94D4
G = < a,b,c,d | a18=c4=d2=1, b2=a9, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 576 in 141 conjugacy classes, 54 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C18, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4×D4, Dic9, Dic9, C36, D18, D18, C2×C18, C2×C18, C2×C18, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C4×D9, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, C22×D9, C22×C18, Dic34D4, C4×Dic9, Dic9⋊C4, D18⋊C4, C9×C22⋊C4, C2×C4×D9, C22×Dic9, C2×C9⋊D4, Dic94D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, D9, C4×S3, C22×S3, C4×D4, D18, S3×C2×C4, S3×D4, D42S3, C4×D9, C22×D9, Dic34D4, C2×C4×D9, D4×D9, D42D9, Dic94D4

Smallest permutation representation of Dic94D4
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)
(1 143 10 134)(2 142 11 133)(3 141 12 132)(4 140 13 131)(5 139 14 130)(6 138 15 129)(7 137 16 128)(8 136 17 127)(9 135 18 144)(19 117 28 126)(20 116 29 125)(21 115 30 124)(22 114 31 123)(23 113 32 122)(24 112 33 121)(25 111 34 120)(26 110 35 119)(27 109 36 118)(37 77 46 86)(38 76 47 85)(39 75 48 84)(40 74 49 83)(41 73 50 82)(42 90 51 81)(43 89 52 80)(44 88 53 79)(45 87 54 78)(55 105 64 96)(56 104 65 95)(57 103 66 94)(58 102 67 93)(59 101 68 92)(60 100 69 91)(61 99 70 108)(62 98 71 107)(63 97 72 106)
(1 43 34 107)(2 42 35 106)(3 41 36 105)(4 40 19 104)(5 39 20 103)(6 38 21 102)(7 37 22 101)(8 54 23 100)(9 53 24 99)(10 52 25 98)(11 51 26 97)(12 50 27 96)(13 49 28 95)(14 48 29 94)(15 47 30 93)(16 46 31 92)(17 45 32 91)(18 44 33 108)(55 132 82 109)(56 131 83 126)(57 130 84 125)(58 129 85 124)(59 128 86 123)(60 127 87 122)(61 144 88 121)(62 143 89 120)(63 142 90 119)(64 141 73 118)(65 140 74 117)(66 139 75 116)(67 138 76 115)(68 137 77 114)(69 136 78 113)(70 135 79 112)(71 134 80 111)(72 133 81 110)
(1 107)(2 108)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 122)(74 123)(75 124)(76 125)(77 126)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)

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

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

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

60 conjugacy classes

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

60 irreducible representations

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

Matrix representation of Dic94D4 in GL4(𝔽37) generated by

 11 31 0 0 6 17 0 0 0 0 1 0 0 0 0 1
,
 9 1 0 0 29 28 0 0 0 0 36 0 0 0 0 36
,
 20 31 0 0 11 17 0 0 0 0 22 3 0 0 11 15
,
 1 0 0 0 0 1 0 0 0 0 22 3 0 0 24 15
G:=sub<GL(4,GF(37))| [11,6,0,0,31,17,0,0,0,0,1,0,0,0,0,1],[9,29,0,0,1,28,0,0,0,0,36,0,0,0,0,36],[20,11,0,0,31,17,0,0,0,0,22,11,0,0,3,15],[1,0,0,0,0,1,0,0,0,0,22,24,0,0,3,15] >;

Dic94D4 in GAP, Magma, Sage, TeX

{\rm Dic}_9\rtimes_4D_4
% in TeX

G:=Group("Dic9:4D4");
// GroupNames label

G:=SmallGroup(288,91);
// by ID

G=gap.SmallGroup(288,91);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽