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G = Dic9⋊C4order 144 = 24·32

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic9⋊C4
G = < a,b,c | a18=c4=1, b2=a9, bab-1=a-1, ac=ca, cbc-1=a9b >

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

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

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

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

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 D6 D9 Dic6 C4×S3 C3⋊D4 D18 Dic18 C4×D9 C9⋊D4 kernel Dic9⋊C4 C2×Dic9 C2×C36 Dic9 C2×C12 C18 C18 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 4 1 1 1 1 3 2 2 2 3 6 6 6

Matrix representation of Dic9⋊C4 in GL4(𝔽37) generated by

 26 0 0 0 24 10 0 0 0 0 11 31 0 0 6 17
,
 14 3 0 0 9 23 0 0 0 0 7 30 0 0 23 30
,
 6 0 0 0 0 6 0 0 0 0 32 27 0 0 10 5
G:=sub<GL(4,GF(37))| [26,24,0,0,0,10,0,0,0,0,11,6,0,0,31,17],[14,9,0,0,3,23,0,0,0,0,7,23,0,0,30,30],[6,0,0,0,0,6,0,0,0,0,32,10,0,0,27,5] >;

Dic9⋊C4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(144,12);
// by ID

G=gap.SmallGroup(144,12);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,2404,208,3461]);
// Polycyclic

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

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