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## G = C4⋊Dic9order 144 = 24·32

### The semidirect product of C4 and Dic9 acting via Dic9/C18=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊Dic9
G = < a,b,c | a4=b18=1, c2=b9, ab=ba, cac-1=a-1, cbc-1=b-1 >

Smallest permutation representation of C4⋊Dic9
Regular action on 144 points
Generators in S144
(1 49 20 60)(2 50 21 61)(3 51 22 62)(4 52 23 63)(5 53 24 64)(6 54 25 65)(7 37 26 66)(8 38 27 67)(9 39 28 68)(10 40 29 69)(11 41 30 70)(12 42 31 71)(13 43 32 72)(14 44 33 55)(15 45 34 56)(16 46 35 57)(17 47 36 58)(18 48 19 59)(73 119 94 135)(74 120 95 136)(75 121 96 137)(76 122 97 138)(77 123 98 139)(78 124 99 140)(79 125 100 141)(80 126 101 142)(81 109 102 143)(82 110 103 144)(83 111 104 127)(84 112 105 128)(85 113 106 129)(86 114 107 130)(87 115 108 131)(88 116 91 132)(89 117 92 133)(90 118 93 134)
(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)
(1 103 10 94)(2 102 11 93)(3 101 12 92)(4 100 13 91)(5 99 14 108)(6 98 15 107)(7 97 16 106)(8 96 17 105)(9 95 18 104)(19 83 28 74)(20 82 29 73)(21 81 30 90)(22 80 31 89)(23 79 32 88)(24 78 33 87)(25 77 34 86)(26 76 35 85)(27 75 36 84)(37 122 46 113)(38 121 47 112)(39 120 48 111)(40 119 49 110)(41 118 50 109)(42 117 51 126)(43 116 52 125)(44 115 53 124)(45 114 54 123)(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)| (1,49,20,60)(2,50,21,61)(3,51,22,62)(4,52,23,63)(5,53,24,64)(6,54,25,65)(7,37,26,66)(8,38,27,67)(9,39,28,68)(10,40,29,69)(11,41,30,70)(12,42,31,71)(13,43,32,72)(14,44,33,55)(15,45,34,56)(16,46,35,57)(17,47,36,58)(18,48,19,59)(73,119,94,135)(74,120,95,136)(75,121,96,137)(76,122,97,138)(77,123,98,139)(78,124,99,140)(79,125,100,141)(80,126,101,142)(81,109,102,143)(82,110,103,144)(83,111,104,127)(84,112,105,128)(85,113,106,129)(86,114,107,130)(87,115,108,131)(88,116,91,132)(89,117,92,133)(90,118,93,134), (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), (1,103,10,94)(2,102,11,93)(3,101,12,92)(4,100,13,91)(5,99,14,108)(6,98,15,107)(7,97,16,106)(8,96,17,105)(9,95,18,104)(19,83,28,74)(20,82,29,73)(21,81,30,90)(22,80,31,89)(23,79,32,88)(24,78,33,87)(25,77,34,86)(26,76,35,85)(27,75,36,84)(37,122,46,113)(38,121,47,112)(39,120,48,111)(40,119,49,110)(41,118,50,109)(42,117,51,126)(43,116,52,125)(44,115,53,124)(45,114,54,123)(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( (1,49,20,60)(2,50,21,61)(3,51,22,62)(4,52,23,63)(5,53,24,64)(6,54,25,65)(7,37,26,66)(8,38,27,67)(9,39,28,68)(10,40,29,69)(11,41,30,70)(12,42,31,71)(13,43,32,72)(14,44,33,55)(15,45,34,56)(16,46,35,57)(17,47,36,58)(18,48,19,59)(73,119,94,135)(74,120,95,136)(75,121,96,137)(76,122,97,138)(77,123,98,139)(78,124,99,140)(79,125,100,141)(80,126,101,142)(81,109,102,143)(82,110,103,144)(83,111,104,127)(84,112,105,128)(85,113,106,129)(86,114,107,130)(87,115,108,131)(88,116,91,132)(89,117,92,133)(90,118,93,134), (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), (1,103,10,94)(2,102,11,93)(3,101,12,92)(4,100,13,91)(5,99,14,108)(6,98,15,107)(7,97,16,106)(8,96,17,105)(9,95,18,104)(19,83,28,74)(20,82,29,73)(21,81,30,90)(22,80,31,89)(23,79,32,88)(24,78,33,87)(25,77,34,86)(26,76,35,85)(27,75,36,84)(37,122,46,113)(38,121,47,112)(39,120,48,111)(40,119,49,110)(41,118,50,109)(42,117,51,126)(43,116,52,125)(44,115,53,124)(45,114,54,123)(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([(1,49,20,60),(2,50,21,61),(3,51,22,62),(4,52,23,63),(5,53,24,64),(6,54,25,65),(7,37,26,66),(8,38,27,67),(9,39,28,68),(10,40,29,69),(11,41,30,70),(12,42,31,71),(13,43,32,72),(14,44,33,55),(15,45,34,56),(16,46,35,57),(17,47,36,58),(18,48,19,59),(73,119,94,135),(74,120,95,136),(75,121,96,137),(76,122,97,138),(77,123,98,139),(78,124,99,140),(79,125,100,141),(80,126,101,142),(81,109,102,143),(82,110,103,144),(83,111,104,127),(84,112,105,128),(85,113,106,129),(86,114,107,130),(87,115,108,131),(88,116,91,132),(89,117,92,133),(90,118,93,134)], [(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)], [(1,103,10,94),(2,102,11,93),(3,101,12,92),(4,100,13,91),(5,99,14,108),(6,98,15,107),(7,97,16,106),(8,96,17,105),(9,95,18,104),(19,83,28,74),(20,82,29,73),(21,81,30,90),(22,80,31,89),(23,79,32,88),(24,78,33,87),(25,77,34,86),(26,76,35,85),(27,75,36,84),(37,122,46,113),(38,121,47,112),(39,120,48,111),(40,119,49,110),(41,118,50,109),(42,117,51,126),(43,116,52,125),(44,115,53,124),(45,114,54,123),(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)])

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + + - + - + - + image C1 C2 C2 C4 S3 D4 Q8 Dic3 D6 D9 Dic6 D12 Dic9 D18 Dic18 D36 kernel C4⋊Dic9 C2×Dic9 C2×C36 C36 C2×C12 C18 C18 C12 C2×C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 4 1 1 1 2 1 3 2 2 6 3 6 6

Matrix representation of C4⋊Dic9 in GL3(𝔽37) generated by

 1 0 0 0 31 0 0 0 6
,
 36 0 0 0 33 0 0 0 9
,
 31 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(37))| [1,0,0,0,31,0,0,0,6],[36,0,0,0,33,0,0,0,9],[31,0,0,0,0,1,0,1,0] >;

C4⋊Dic9 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_9
% in TeX

G:=Group("C4:Dic9");
// GroupNames label

G:=SmallGroup(144,13);
// by ID

G=gap.SmallGroup(144,13);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,55,2404,208,3461]);
// Polycyclic

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

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