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## G = C3×Q8⋊2D9order 432 = 24·33

### Direct product of C3 and Q8⋊2D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C3×Q8⋊2D9
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C3×D36 — C3×Q8⋊2D9
 Lower central C9 — C18 — C36 — C3×Q8⋊2D9
 Upper central C1 — C6 — C12 — C3×Q8

Generators and relations for C3×Q82D9
G = < a,b,c,d,e | a3=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 294 in 72 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, C12, C12, D6, C2×C6, SD16, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, C3×C9, C36, C36, D18, C3×C12, C3×C12, S3×C6, Q82S3, C3×SD16, C3×D9, C3×C18, C9⋊C8, D36, Q8×C9, Q8×C9, C3×C3⋊C8, C3×D12, Q8×C32, C3×C36, C3×C36, C6×D9, Q82D9, C3×Q82S3, C3×C9⋊C8, C3×D36, Q8×C3×C9, C3×Q82D9
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, D9, C3×S3, C3⋊D4, C3×D4, D18, S3×C6, Q82S3, C3×SD16, C3×D9, C9⋊D4, C3×C3⋊D4, C6×D9, Q82D9, C3×Q82S3, C3×C9⋊D4, C3×Q82D9

Smallest permutation representation of C3×Q82D9
On 144 points
Generators in S144
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)
(1 28 10 19)(2 29 11 20)(3 30 12 21)(4 31 13 22)(5 32 14 23)(6 33 15 24)(7 34 16 25)(8 35 17 26)(9 36 18 27)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)(73 91 82 100)(74 92 83 101)(75 93 84 102)(76 94 85 103)(77 95 86 104)(78 96 87 105)(79 97 88 106)(80 98 89 107)(81 99 90 108)(109 136 118 127)(110 137 119 128)(111 138 120 129)(112 139 121 130)(113 140 122 131)(114 141 123 132)(115 142 124 133)(116 143 125 134)(117 144 126 135)
(1 46 10 37)(2 47 11 38)(3 48 12 39)(4 49 13 40)(5 50 14 41)(6 51 15 42)(7 52 16 43)(8 53 17 44)(9 54 18 45)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)(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 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 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 81)(9 80)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 90)(18 89)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 99)(27 98)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 108)(36 107)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 117)(45 116)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 126)(54 125)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 135)(63 134)(64 142)(65 141)(66 140)(67 139)(68 138)(69 137)(70 136)(71 144)(72 143)

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

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

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

81 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 9A ··· 9I 12A 12B 12C ··· 12M 18A ··· 18I 24A 24B 24C 24D 36A ··· 36AA order 1 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 8 8 9 ··· 9 12 12 12 ··· 12 18 ··· 18 24 24 24 24 36 ··· 36 size 1 1 36 1 1 2 2 2 2 4 1 1 2 2 2 36 36 18 18 2 ··· 2 2 2 4 ··· 4 2 ··· 2 18 18 18 18 4 ··· 4

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 SD16 D9 C3×S3 C3×D4 C3⋊D4 D18 S3×C6 C3×SD16 C3×D9 C9⋊D4 C3×C3⋊D4 C6×D9 C3×C9⋊D4 Q8⋊2S3 Q8⋊2D9 C3×Q8⋊2S3 C3×Q8⋊2D9 kernel C3×Q8⋊2D9 C3×C9⋊C8 C3×D36 Q8×C3×C9 Q8⋊2D9 C9⋊C8 D36 Q8×C9 Q8×C32 C3×C18 C3×C12 C3×C9 C3×Q8 C3×Q8 C18 C3×C6 C12 C12 C9 Q8 C6 C6 C4 C2 C32 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 3 2 2 2 3 2 4 6 6 4 6 12 1 3 2 6

Matrix representation of C3×Q82D9 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 48 0 0 0 0 3 72
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 71 14 0 0 0 0 57 2
,
 16 0 0 0 0 0 0 32 0 0 0 0 0 0 58 70 0 0 0 0 46 14 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 41 0 0 0 0 57 0 0 0 0 0 0 0 1 53 0 0 0 0 0 72 0 0 0 0 0 0 33 59 0 0 0 0 36 40

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,48,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,71,57,0,0,0,0,14,2],[16,0,0,0,0,0,0,32,0,0,0,0,0,0,58,46,0,0,0,0,70,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,57,0,0,0,0,41,0,0,0,0,0,0,0,1,0,0,0,0,0,53,72,0,0,0,0,0,0,33,36,0,0,0,0,59,40] >;

C3×Q82D9 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_2D_9
% in TeX

G:=Group("C3xQ8:2D9");
// GroupNames label

G:=SmallGroup(432,157);
// by ID

G=gap.SmallGroup(432,157);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=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,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽