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## G = C2×Q8⋊3D9order 288 = 25·32

### Direct product of C2 and Q8⋊3D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×Q8⋊3D9
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C2×C4×D9 — C2×Q8⋊3D9
 Lower central C9 — C18 — C2×Q8⋊3D9
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q83D9
G = < a,b,c,d,e | a2=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1032 in 246 conjugacy classes, 108 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C9, Dic3, C12, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, D9, C18, C18, C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C2×C4○D4, Dic9, C36, D18, D18, C2×C18, S3×C2×C4, C2×D12, Q83S3, C6×Q8, C4×D9, D36, C2×Dic9, C2×C36, Q8×C9, C22×D9, C2×Q83S3, C2×C4×D9, C2×D36, Q83D9, Q8×C18, C2×Q83D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, D9, C22×S3, C2×C4○D4, D18, Q83S3, S3×C23, C22×D9, C2×Q83S3, Q83D9, C23×D9, C2×Q83D9

Smallest permutation representation of C2×Q83D9
On 144 points
Generators in S144
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 64 10 55)(2 65 11 56)(3 66 12 57)(4 67 13 58)(5 68 14 59)(6 69 15 60)(7 70 16 61)(8 71 17 62)(9 72 18 63)(19 37 28 46)(20 38 29 47)(21 39 30 48)(22 40 31 49)(23 41 32 50)(24 42 33 51)(25 43 34 52)(26 44 35 53)(27 45 36 54)(73 136 82 127)(74 137 83 128)(75 138 84 129)(76 139 85 130)(77 140 86 131)(78 141 87 132)(79 142 88 133)(80 143 89 134)(81 144 90 135)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)
(1 109 10 118)(2 110 11 119)(3 111 12 120)(4 112 13 121)(5 113 14 122)(6 114 15 123)(7 115 16 124)(8 116 17 125)(9 117 18 126)(19 127 28 136)(20 128 29 137)(21 129 30 138)(22 130 31 139)(23 131 32 140)(24 132 33 141)(25 133 34 142)(26 134 35 143)(27 135 36 144)(37 82 46 73)(38 83 47 74)(39 84 48 75)(40 85 49 76)(41 86 50 77)(42 87 51 78)(43 88 52 79)(44 89 53 80)(45 90 54 81)(55 100 64 91)(56 101 65 92)(57 102 66 93)(58 103 67 94)(59 104 68 95)(60 105 69 96)(61 106 70 97)(62 107 71 98)(63 108 72 99)
(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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(73 135)(74 134)(75 133)(76 132)(77 131)(78 130)(79 129)(80 128)(81 127)(82 144)(83 143)(84 142)(85 141)(86 140)(87 139)(88 138)(89 137)(90 136)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3 4A ··· 4F 4G 4H 4I 4J 6A 6B 6C 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 2 ··· 2 3 4 ··· 4 4 4 4 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 18 ··· 18 2 2 ··· 2 9 9 9 9 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D6 D6 C4○D4 D9 D18 D18 Q8⋊3S3 Q8⋊3D9 kernel C2×Q8⋊3D9 C2×C4×D9 C2×D36 Q8⋊3D9 Q8×C18 C6×Q8 C2×C12 C3×Q8 C18 C2×Q8 C2×C4 Q8 C6 C2 # reps 1 3 3 8 1 1 3 4 4 3 9 12 2 6

Matrix representation of C2×Q83D9 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 36 0 0 0 0 36 0 0 0 0 29 12 0 0 10 8
,
 1 0 0 0 0 1 0 0 0 0 11 2 0 0 13 26
,
 6 26 0 0 11 17 0 0 0 0 1 0 0 0 0 1
,
 31 11 0 0 17 6 0 0 0 0 29 12 0 0 4 8
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,29,10,0,0,12,8],[1,0,0,0,0,1,0,0,0,0,11,13,0,0,2,26],[6,11,0,0,26,17,0,0,0,0,1,0,0,0,0,1],[31,17,0,0,11,6,0,0,0,0,29,4,0,0,12,8] >;

C2×Q83D9 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3D_9
% in TeX

G:=Group("C2xQ8:3D9");
// GroupNames label

G:=SmallGroup(288,360);
// by ID

G=gap.SmallGroup(288,360);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,185,80,6725,292,9414]);
// Polycyclic

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

׿
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𝔽