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## G = D4.Dic9order 288 = 25·32

### The non-split extension by D4 of Dic9 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4.Dic9
 Chief series C1 — C3 — C9 — C18 — C36 — C9⋊C8 — C2×C9⋊C8 — D4.Dic9
 Lower central C9 — C18 — D4.Dic9
 Upper central C1 — C4 — C4○D4

Generators and relations for D4.Dic9
G = < a,b,c,d | a4=b2=1, c18=a2, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c17 >

Subgroups: 212 in 93 conjugacy classes, 62 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, C9, C12, C12, C2×C6, C2×C8, M4(2), C4○D4, C18, C18, C3⋊C8, C2×C12, C3×D4, C3×Q8, C8○D4, C36, C36, C2×C18, C2×C3⋊C8, C4.Dic3, C3×C4○D4, C9⋊C8, C9⋊C8, C2×C36, D4×C9, Q8×C9, D4.Dic3, C2×C9⋊C8, C4.Dic9, C9×C4○D4, D4.Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, D9, C2×Dic3, C22×S3, C8○D4, Dic9, D18, C22×Dic3, C2×Dic9, C22×D9, D4.Dic3, C22×Dic9, D4.Dic9

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 8C 8D 8E ··· 8J 9A 9B 9C 12A 12B 12C 12D 12E 18A 18B 18C 18D ··· 18L 36A ··· 36F 36G ··· 36O order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 6 8 8 8 8 8 ··· 8 9 9 9 12 12 12 12 12 18 18 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 2 2 2 2 1 1 2 2 2 2 4 4 4 9 9 9 9 18 ··· 18 2 2 2 2 2 4 4 4 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - - + + - - image C1 C2 C2 C2 C4 C4 S3 D6 Dic3 Dic3 D9 C8○D4 D18 Dic9 Dic9 D4.Dic3 D4.Dic9 kernel D4.Dic9 C2×C9⋊C8 C4.Dic9 C9×C4○D4 D4×C9 Q8×C9 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C9 C2×C4 D4 Q8 C3 C1 # reps 1 3 3 1 6 2 1 3 3 1 3 4 9 9 3 2 6

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

 0 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 46 0 0 0 0 46 0 0 0 0 3 28 0 0 45 31
,
 51 0 0 0 0 51 0 0 0 0 19 59 0 0 5 54
G:=sub<GL(4,GF(73))| [0,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,0,46,0,0,0,0,3,45,0,0,28,31],[51,0,0,0,0,51,0,0,0,0,19,5,0,0,59,54] >;

D4.Dic9 in GAP, Magma, Sage, TeX

D_4.{\rm Dic}_9
% in TeX

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

G:=SmallGroup(288,158);
// by ID

G=gap.SmallGroup(288,158);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,219,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^18=a^2,d^2=c^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^17>;
// generators/relations

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