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## G = Dic9.D4order 288 = 25·32

### 1st non-split extension by Dic9 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — Dic9.D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — D18⋊C4 — Dic9.D4
 Lower central C9 — C2×C18 — Dic9.D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic9.D4
G = < a,b,c,d | a18=c4=1, b2=d2=a9, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a9b, dcd-1=a9c-1 >

Subgroups: 524 in 114 conjugacy classes, 40 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C22, C22 [×6], S3, C6 [×3], C6, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×2], C23, C23, C9, Dic3 [×4], C12 [×2], D6 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4, C22⋊C4 [×3], C2×D4, C2×Q8, D9, C18 [×3], C18, Dic6 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C4.4D4, Dic9 [×2], Dic9 [×2], C36 [×2], D18 [×3], C2×C18, C2×C18 [×3], C4×Dic3, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, Dic18 [×2], C2×Dic9 [×3], C9⋊D4 [×2], C2×C36 [×2], C22×D9, C22×C18, C23.11D6, C4×Dic9, D18⋊C4 [×2], C18.D4, C9×C22⋊C4, C2×Dic18, C2×C9⋊D4, Dic9.D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C22×S3, C4.4D4, D18 [×3], C4○D12, S3×D4, D42S3, C22×D9, C23.11D6, D365C2, D4×D9, D42D9, 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 64 10 55)(2 63 11 72)(3 62 12 71)(4 61 13 70)(5 60 14 69)(6 59 15 68)(7 58 16 67)(8 57 17 66)(9 56 18 65)(19 52 28 43)(20 51 29 42)(21 50 30 41)(22 49 31 40)(23 48 32 39)(24 47 33 38)(25 46 34 37)(26 45 35 54)(27 44 36 53)(73 139 82 130)(74 138 83 129)(75 137 84 128)(76 136 85 127)(77 135 86 144)(78 134 87 143)(79 133 88 142)(80 132 89 141)(81 131 90 140)(91 121 100 112)(92 120 101 111)(93 119 102 110)(94 118 103 109)(95 117 104 126)(96 116 105 125)(97 115 106 124)(98 114 107 123)(99 113 108 122)
(1 25 96 90)(2 26 97 73)(3 27 98 74)(4 28 99 75)(5 29 100 76)(6 30 101 77)(7 31 102 78)(8 32 103 79)(9 33 104 80)(10 34 105 81)(11 35 106 82)(12 36 107 83)(13 19 108 84)(14 20 91 85)(15 21 92 86)(16 22 93 87)(17 23 94 88)(18 24 95 89)(37 125 131 55)(38 126 132 56)(39 109 133 57)(40 110 134 58)(41 111 135 59)(42 112 136 60)(43 113 137 61)(44 114 138 62)(45 115 139 63)(46 116 140 64)(47 117 141 65)(48 118 142 66)(49 119 143 67)(50 120 144 68)(51 121 127 69)(52 122 128 70)(53 123 129 71)(54 124 130 72)
(1 81 10 90)(2 80 11 89)(3 79 12 88)(4 78 13 87)(5 77 14 86)(6 76 15 85)(7 75 16 84)(8 74 17 83)(9 73 18 82)(19 102 28 93)(20 101 29 92)(21 100 30 91)(22 99 31 108)(23 98 32 107)(24 97 33 106)(25 96 34 105)(26 95 35 104)(27 94 36 103)(37 116 46 125)(38 115 47 124)(39 114 48 123)(40 113 49 122)(41 112 50 121)(42 111 51 120)(43 110 52 119)(44 109 53 118)(45 126 54 117)(55 131 64 140)(56 130 65 139)(57 129 66 138)(58 128 67 137)(59 127 68 136)(60 144 69 135)(61 143 70 134)(62 142 71 133)(63 141 72 132)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 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 3 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 4 36 2 2 2 4 18 18 18 18 36 2 2 2 4 4 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

54 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 C2 S3 D4 D6 D6 C4○D4 D9 D18 D18 C4○D12 D36⋊5C2 S3×D4 D4⋊2S3 D4×D9 D4⋊2D9 kernel Dic9.D4 C4×Dic9 D18⋊C4 C18.D4 C9×C22⋊C4 C2×Dic18 C2×C9⋊D4 C3×C22⋊C4 Dic9 C2×C12 C22×C6 C18 C22⋊C4 C2×C4 C23 C6 C2 C6 C6 C2 C2 # reps 1 1 2 1 1 1 1 1 2 2 1 4 3 6 3 4 12 1 1 3 3

Matrix representation of Dic9.D4 in GL6(𝔽37)

 17 11 0 0 0 0 26 6 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 36 36 0 0 0 0 0 0 0 36 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 35 36
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 6 0 0 0 0 31 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 36 0 0 0 0 0 1 1 0 0 0 0 0 0 0 6 0 0 0 0 6 0 0 0 0 0 0 0 6 0 0 0 0 0 25 31

G:=sub<GL(6,GF(37))| [17,26,0,0,0,0,11,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,36,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,36,0,0,0,0,0,0,0,1,35,0,0,0,0,1,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,31,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[36,1,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,6,0,0,0,0,0,0,0,6,25,0,0,0,0,0,31] >;

Dic9.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_9.D_4
% in TeX

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

G:=SmallGroup(288,95);
// by ID

G=gap.SmallGroup(288,95);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,219,6725,292,9414]);
// Polycyclic

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

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