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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 [×3], C3, C4, C4 [×3], C22 [×3], C6, C6 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C9, C12, C12 [×3], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C18, C18 [×3], C3⋊C8 [×4], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C36, C36 [×3], C2×C18 [×3], C2×C3⋊C8 [×3], C4.Dic3 [×3], C3×C4○D4, C9⋊C8, C9⋊C8 [×3], C2×C36 [×3], D4×C9 [×3], Q8×C9, D4.Dic3, C2×C9⋊C8 [×3], C4.Dic9 [×3], C9×C4○D4, D4.Dic9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, D9, C2×Dic3 [×6], C22×S3, C8○D4, Dic9 [×4], D18 [×3], C22×Dic3, C2×Dic9 [×6], C22×D9, D4.Dic3, C22×Dic9, D4.Dic9

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

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

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

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

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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